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Geometrization of Three-Dimensional Orbifolds via Ricci Flow

A three-dimensional closed orientable orbifold (with no bad suborbifolds) is known to have a geometric decomposition from work of Perelman along with earlier work of Boileau-Leeb-Porti and Cooper-Hodgson-Kerckhoff. We give a new, logically independent, unified proof of the geometrization of orbifolds, using Ricci flow. Along the way we develop some tools for the geometry of orbifolds that may be of independent interest.

Thurston's geometrization conjecture for 3-manifolds states that every closed orientable 3-manifold has a canonical decomposition into geometric pieces. In the early 1980's Thurston announced a proof of the conjecture for Haken manifolds [57], with written proofs appearing much later [34,39,45,46]. The conjecture was settled completely a few years ago by Perelman in his spectacular work using Hamilton's Ricci flow [47,48].
Thurston also formulated a geometrization conjecture for orbifolds. We recall that orbifolds are similar to manifolds, except that they are locally modelled on quotients of the form R n /G, where G ⊂ O(n) is a finite subgroup of the orthogonal group. Although the terminology is relatively recent, orbifolds have a long history in mathematics, going back to the classification of crystallographic groups and Fuchsian groups. In this paper, using Ricci flow, we will give a new proof of the geometrization conjecture for orbifolds: Theorem 1.1. Let O be a closed connected orientable three-dimensional orbifold which does not contain any bad embedded 2-dimensional suborbifolds. Then O has a geometric decomposition.
The existing proof of Theorem 1.1 is based on a canonical splitting of O along spherical and Euclidean 2-dimensional suborbifolds, which is analogous to the prime and JSJ decomposition of 3-manifolds. This splitting reduces Theorem 1.1 to two separate cases -when O is a manifold, and when O has a nonempty singular locus and satisfies an irreducibility condition. The first case is Perelman's theorem for manifolds. Thurston announced a proof of the latter case in [56] and gave an outline. The first complete proofs of Theorem 1.1 are due to Boileau-Leeb-Porti [4] and Cooper-Hodgson-Kerckhoff [17]. The monographs [5,17] give excellent expositions of 3-orbifolds and their geometrization.
1.2. Discussion of the proof. The main purpose of this paper is to provide a new proof of Theorem 1.1. Our proof is an extension of Perelman's proof of geometrization for 3manifolds to orbifolds, bypassing [4,17]. The motivation for this alternate approach is twofold. First, anyone interested in the geometrization of general orbifolds as in Theorem 1.1 will necessarily have to go through Perelman's Ricci flow proof in the manifold case, and also absorb foundational results about orbifolds. At that point, the additional effort required to deal with general orbifolds is relatively minor in comparison with the original proofs in [4,17]. These latter proofs involve a number of ingredients, including Thurston's geometrization of Haken manifolds, the deformation and collapsing theory of hyperbolic cone manifolds, and some Alexandrov space theory. Also, in contrast to the existing proof of Theorem 1.1, the Ricci flow argument gives a unified approach to geometrization for both manifolds and orbifolds.
Many of the steps in Perelman's proof have evident orbifold generalizations, whereas some do not. It would be unwieldy to rewrite all the details of Perelman's proof, on the level of [35], while making notational changes from manifolds to orbifolds. Consequently, we focus on the steps in Perelman's proof where an orbifold extension is not immediate. For a step where the orbifold extension is routine, we make the precise orbifold statement and indicate where the analogous manifold proof occurs in [35].
In the course of proving Theorem 1.1, we needed to develop a number of foundational results about the geometry of orbifolds. Some of these may be of independent interest, or of use for subsequent work in this area, such as the compactness theorem for Riemannian orbifolds, critical point theory, and the soul theorem.
Let us mention one of the steps where the orbifold extension could a priori be an issue. This is where one characterizes the topology of the thin part of the large-time orbifold. To do this, one first needs a sufficiently flexible proof in the manifold case. We provided such a proof in [36]. The proof in [36] uses some basic techniques from Alexandrov geometry, combined with smoothness results in appropriate places. It provides a decomposition of the thin part into various pieces which together give an explicit realization of the thin part as a graph manifold. When combined with preliminary results that are proved in this paper, we can extend the techniques of [36] to orbifolds. We get a decomposition of the thin part of the large-time orbifold into various pieces, similar to those in [36]. We show that these pieces give an explicit realization of each component of the thin part as either a graph orbifold or one of a few exceptional cases. This is more involved to prove in the orbifold case than in the manifold case but the basic strategy is the same.
1.3. Organization of the paper. The structure of this paper is as follows. One of our tasks is to provide a framework for the topology and Riemannian geometry of orbifolds, so that results about Ricci flow on manifolds extend as easily as possible to orbifolds. In Section 2 we recall the relevant notions that we need from orbifold topology. We then introduce Riemannian orbifolds and prove the orbifold versions of some basic results from Riemannian geometry, such as the de Rham decomposition and critical point theory.
Section 3 is concerned with noncompact nonnegatively curved orbifolds. We prove the orbifold version of the Cheeger-Gromoll soul theorem. We list the diffeomorphism types of noncompact nonnegatively curved orbifolds with dimension at most three.
In Section 4 we prove a compactness theorem for Riemannian orbifolds. Section 5 contains some preliminary information about Ricci flow on orbifolds, along with the classification of the diffeomorphism types of compact nonnegatively curved three-dimensional orbifolds. We also show how to extend Perelman's no local collapsing theorem to orbifolds. Section 6 is devoted to κ-solutions. Starting in Section 7, we specialize to three-dimensional orientable orbifolds with no bad 2-dimensional suborbifolds. We show how to extend Perelman's results in order to construct a Ricci flow with surgery.
In Section 8 we show that the thick part of the large-time geometry approaches a finitevolume orbifold of constant negative curvature. Section 9 contains the topological characterization of the thin part of the large-time geometry.
Section 10 concerns the incompressibility of hyperbolic cross-sections. Rather than using minimal disk techniques as initiated by Hamilton [32], we follow an approach introduced by Perelman [48, Section 8] that uses a monotonic quantity, as modified in [35,Section 93.4].
The appendix contains topological facts about graph orbifolds. We show that a "weak" graph orbifold has a connected sum decomposition in terms of "strong" graph orbifolds. This material is probably known to some experts but we were unable to find references in the literature, so we include complete proofs.
After writing this paper we learned that Daniel Faessler independently proved Proposition 9.7, which is the orbifold version of the collapsing theorem [22]. 1.4. Acknowledgements. We thank Misha Kapovich and Sylvain Maillot for orbidiscussions.

Orbifold topology and geometry
In this section we first review the differential topology of orbifolds. Subsections 2.1 and 2.2 contain information about orbifolds in any dimension. In some cases we give precise definitions and in other cases we just recall salient properties, referring to the monographs [5,17] for more detailed information. Subsections 2.3 and 2.4 are concerned with lowdimensional orbifolds.
We then give a short exposition of aspects of the differential geometry of orbifolds, in Subsection 2.5. It is hard to find a comprehensive reference for this material and so we flag the relevant notions; see [6] for further discussion of some points. Subsection 2.6 shows how to do critical point theory on orbifolds. Subsection 2.7 discusses the smoothing of functions on orbifolds.
For notation, B n is the open unit n-ball, D n is the closed unit n-ball and I = [−1, 1]. We let D k denote the dihedral group of order 2k. V → V ′ and a homomorphism h : G → G ′ so that for all g ∈ G, ρ ′ (h(g)) • T = T • ρ(g). The linear map is injective (resp. surjective) if T is injective (resp. surjective) and h is injective (resp. surjective).
A local model is a pair ( U , G), where U is a connected open subset of a Euclidean space and G is a finite group that acts smoothly and effectively on U, on the right. (Effectiveness means that the homomorphism G → Diff( U ) is injective.) We will sometimes write U for U /G, endowed with the quotient topology.
A smooth map between local models ( U 1 , G 1 ) and ( U 2 , G 2 ) is given by a smooth map f : U 1 → U 2 and a homomorphism ρ : G 1 → G 2 so that f is ρ-equivariant, i.e. f (xg 1 ) = f (x)ρ(g 1 ). We do not assume that ρ is injective or surjective. The map between local models is an embedding if f is an embedding; it follows from effectiveness that ρ is injective in this case.
A smooth map f : O 1 → O 2 between orbifolds is given by a continuous map |f | : |O 1 | → |O 2 | with the property that for each p ∈ |O 1 |, there are • Local models ( U 1 , G 1 ) and ( U 2 , G 2 ) for p and f (p), respectively, and • A smooth map f : ( U 1 , G 1 ) → ( U 2 , G 2 ) between local models so that the diagram commutes.
There is an induced homomorphism from G p to G f (p) . We emphasize that to define a smooth map f between two orbifolds, one must first define a map |f | between their underlyihg spaces.
We write C ∞ (O) for the smooth maps f : O → R.
A smooth map f : If a discrete group Γ acts properly discontinuously on a manifold M then there is a quotient orbifold, which we denote by M//Γ. It has |M//Γ| = M/Γ. Hence if O is an orbifold and ( U, G) is a local model for O then we can say that O U is diffeomorphic to U //G. An orbifold O is good if O = M//Γ for some manifold M and some discrete group Γ. It is very good if Γ can be taken to be finite. A bad orbifold is one that is not good.
Similarly, suppose that a discrete group Γ acts by diffeomorphisms on an orbifold O. We say that it acts properly discontinuously if the action of Γ on |O| is properly discontinuous. Then there is a quotient orbifold O//Γ, with |O//Γ| = |O|/Γ; see Remark 2.10.
An orbifiber bundle consists of a smooth map π : O 1 → O 2 between two orbifolds, along with a third orbifold O 3 such that • |π| is surjective, and • For each p ∈ |O 2 |, there is a local model ( U , G p ) around p along with an action of (Note that if O 2 is a manifold then the orbifiber bundle π : O 1 → O 2 has a local product structure.) The fiber of the orbifiber bundle is O 3 . Note that for p 1 ∈ |O 1 |, the homomorphism G p 1 → G |π|(p 1 ) is surjective.
A section of an orbifiber bundle π : O 1 → O 2 is a smooth map s : O 2 → O 1 such that π • s is the identity on O 2 .
A covering map π : O 1 → O 2 is a orbifiber bundle with a zero-dimensional fiber. Given p 2 ∈ |O 2 | and p 1 ∈ |π| −1 (p 2 ), there are a local model ( U , G 2 ) around p 2 and a subgroup G 2 ⊂ G 1 so that ( U , G 1 ) is a local model around p 1 and the map π is locally ( U , G 1 ) → ( U , G 2 ).
A rank-m orbivector bundle V → O over O is locally isomorphic to (R m × U )/G p , where G p acts linearly on R m .
The tangent bundle T O of an orbifold O is an orbivector bundle which is locally diffeomorphic to T U α //G α . Given p ∈ |O|, if p ∈ U covers p then the tangent space T p O is isomorphic to the orbivector space (T p U , G p ). The tangent cone at p is C p |O| ∼ = T p U/G p .
A smooth vector field V is a smooth section of T O. In terms of a local model ( U, G), the vector field V restricts to a vector field on U which is G-invariant.
A smooth map f : O 1 → O 2 gives rise to the differential, a orbivector bundle map df : T O 1 → T O 2 . At a point p ∈ |O|, in terms of local models we have a map f : ( U 1 , G 1 ) → ( U 2 , G 2 ) which gives rise to a G p -equivariant map d f p : T p U 1 → T f ( p) U 2 and hence to a linear map df p : Given a smooth map f : O 1 → O 2 and a point p ∈ |O 1 |, we say that f is a submersion at p (resp. immersion at p) if the map df p : T p O 1 → T |f |(p) O 2 is surjective (resp. injective).
3. If f is a submersion at p then there is an orbifold O 3 on which G |f |(p) acts, along with a local model ( U , G |f |(p) ) around |f |(p), so that f is equivalent near p to the projection map (O 3 × U)//G |f |(p) → U//G |f |(p) .
We say that f : O 1 → O 2 is a submersion (resp. immersion) if it is a submersion (resp. immersion) at p for all p ∈ |O 1 |. We will sketch a proof of Lemma 2.4 in Remark 2.12.
In particular, a proper surjective local diffeomorphism to a connected orbifold is a covering map with finite fibers.
An immersion f : O 1 → O 2 has a normal bundle NO 1 → O 1 whose fibers have the following local description. Given p ∈ |O 1 |, let f be described in terms of local models ( U 1 , G p ) and ( U 2 , G |f |(p) ) by a ρ-equivariant immersion f : U 1 → U 2 . Let F p ⊂ G |f |(p) be the subgroup which fixes Im(d f p ). Then the normal space N p O 1 is the orbivector space morphically to its image in |O|. From effectiveness, for each p ∈ |O ′ |, the homomorphism ρ : G p → G |f |(p) is injective. There is a neighborhood of O ′ which is diffeomorphic to the normal bundle NO ′ .
If O ′ is a codimension-1 suborbifold of O then we say that O ′ is two-sided if the normal bundle NO ′ has a nowhere-zero section.
We can talk about two suborbifolds meeting transversely, as defined using local models. Note, however, that we cannot always perform an isotopy to make one suborbifold transverse to another one.
Let O be an oriented orbifold (possibly disconnected). Let D 1 and D 2 be disjoint codimension-zero suborbifolds-with-boundary, both oriented-diffeomorphic to D n //Γ. Then the operation of performing 0-surgery along D 1 , D 2 produces the new oriented orbifold In the manifold case, a connected sum is the same thing as a 0-surgery along a pair {D 1 , D 2 } which lie in different connected components of O. Note that unlike in the manifold case, O ′ is generally not uniquely determined up to diffeomorphism by knowing the connected components containing D 1 and D 2 . For example, even if O is connected, D 1 and D 2 may or may not lie on the same connected component of the singular set.
If O 1 and O 2 are oriented orbifolds, with D 1 ⊂ O 1 and D 2 ⊂ O 2 both oriented diffeomorphic to D n //Γ, then we may write O 1 # S n−1 //Γ O 2 for the connected sum. This notation is slightly ambiguous since the location of D 1 and D 2 is implicit. We will write O# S n−1 //Γ to denote a 0-surgery on a single orbifold O. Again the notation is slightly ambiguous, since the location of Lemma 2.5. Given an involutive distribution E on O, for any p ∈ |O| there is a unique maximal suborbifold passing through p which is tangent to E.
Orbifolds have partitions of unity. Lemma 2. 6. Given an open cover {U α } α∈A of |O|, there is a collection of functions ρ α ∈ C ∞ (O) such that Proof. The proof is similar to the manifold case, using local models ( U , G) consisting of coordinate neighborhoods, along with compactly supported G-invariant smooth functions on U .
A curve in an orbifold is a smooth map γ : I → O defined on an interval I ⊂ R. A loop is a curve γ with |γ|(0) = |γ|(1) ∈ |O|.

2.2.
Universal cover and fundamental group. We follow the presentation in [ 1]. An elementary homotopy between two special curves is a smooth homotopy of γ in U, relative to γ(a) and γ(b). A homotopy of γ is what's generated by elementary homotopies.
If O is connected then the universal cover O of O can be constructed as the set of special curves starting at p, modulo homotopy. It has a natural orbifold structure. The fundamental group π 1 (O, p) is given by special loops (i.e. special curves γ with |γ(1)| = p) modulo homotopy. Up to isomorphism, π 1 (O, p) is independent of the choice of p.
If O is connected and a discrete group Γ acts properly discontinuously on O then there is a short exact sequence Remark 2.8. A more enlightening way to think of an orbifold is to consider it as a smooth properétale groupoid G, as explained in [1,10,42]. Given an orbifold, the unit space G 0 of the corresponding groupoid is the disjoint union of the U α 's. The groupoid elements γ ∈ G come from a point p ∈ |O|, a triple ( U 1 , U 2 , U 3 ) as in part 5. of Definition 2.1 and lifts p ′ 1 ∈ U 1 , p 2 ∈ U 2 of p. Conversely, a smooth properétale groupoid gives rise to an orbifold. Speaking hereafter just of such groupoids, Morita-equivalent groupoids give equivalent orbifolds.
A groupoid morphism gives rise to an orbifold map. Taking into account Morita equivalence, from the groupoid viewpoint the right notion of an orbifold map would be a Hilsum-Skandalis map between groupoids. These turn out to correspond to good maps between orbifolds, as later defined by Chen-Ruan [1]. This is a more restricted class of maps between orbifolds than what we consider. The distinction is that one can pull back orbivector bundles under good maps, but not always under smooth maps in our sense. Orbifold diffeomorphisms in our sense are automatically good maps. For some purposes it would be preferable to only deal with good maps, but for simplicity we will stick with our orbifold definitions.
A Lie groupoid G has a classifying space BG. In the orbifold case, if G is an associated etale groupoid then π 1 (O) ∼ = π 1 (BG). The definition of the latter can be made explicit in terms of paths and homotopies; see [10,27]. In the case of effective orbifolds, the definition is equivalent to the one of the present paper.
More information is in [1,42] and references therein.
2.3. Low-dimensional orbifolds. We list the connected compact boundaryless orbifolds of low dimension. We mostly restrict here to the orientable case. (The nonorientable ones also arise; even if the total space of an orbifiber bundle is orientable, the base may fail to be orientable.) 2.3.1. Zero dimensions. The only possibility is a point.
2.3.2. One dimension. There are two possibilities : S 1 and S 1 //Z 2 . For the latter, the nonzero element of Z 2 acts by complex conjugation on S 1 , and |S 1 //Z 2 | is an interval. Note that S 1 //Z 2 is not orientable.
2.3.3. Two dimensions. For notation, if S is a connected oriented surface then S(k 1 , . . . , k r ) denotes the oriented orbifold O with |O| = S, having singular points of order k 1 , . . . , k r > 1. Any connected oriented 2-orbifold can be written in this way. An orbifold of the form S 2 (p, q, r) is called a turnover.
The other closed orientable 2-orbifolds are hyperbolic.
We will also need some 2-orbifolds with boundary, namely Here the nontrivial element of Z 2 acts by involution on I and by complex conjugation on S 1 . We can also write D 2 (2, 2) as the quotient {z ∈ C : 1 2 ≤ |z| ≤ 2}//Z 2 , where the nontrivial element of Z 2 sends z to z −1 .
• D 2 //Z 2 , where Z 2 acts by complex conjugation on D 2 . Then |∂(D 2 //Z 2 )| is a circle with one orbifold boundary component and one reflector component. See Figure 1, where the dark line indicates the reflector component. • D 2 //D k = D 2 (k)//Z 2 , for k > 1, where D k is the dihedral group and Z 2 acts by complex conjugation on D 2 (k). Then ∂|D 2 //Z 2 | is a circle with one orbifold boundary component, one corner reflector point of order k and two reflector components. See Figure 2.

Three dimensions.
If O is an orientable three-dimensional orbifold then |O| is an orientable topological 3-manifold. The singular locus in |O| is a trivalent graph with each edge labelled by an integer greater than one, under the constraint that if edges with labels  p, q, r meet at a vertex then 1 p + 1 q + 1 r > 1. That is, there is a neighborhood of the vertex which is a cone over an orientable spherical 2-orbifold.
Specifying such a topological 3-manifold and such a labelled trivalent graph is equivalent to specifying an orientable three-dimensional orbifold.
The solid-toric 3-orbifolds are There is no singular locus.
• S 1 × D 2 (k). The singular locus is a core curve in a solid torus. See Figure 5 • S 1 × Z 2 D 2 . The singular locus consists of two arcs in a 3-disk, each labelled by 2.
Given Γ ∈ Isom + (S 2 ), we can consider the quotient S 3 //Γ where Γ acts on S 3 by the suspension of its action on S 2 . That is, we are identifying Isom + (S 2 ) with SO(3) and using the embedding SO(3) → SO(4) to let Γ act on S 3 .
An orientable three-dimensional orbifold O is irreducible if it contains no embedded bad 2-dimensional suborbifolds, and any embedded orientable spherical 2-orbifold S 2 //Γ bounds a discal 3-orbifold D 3 //Γ in O. Figure 8 shows an embedded bad 2-dimensional suborbifold Σ. Figure 9 shows an embedded spherical 2-suborbifold S 2 (k, k) that does not bound a discal 3-orbifold; the shaded regions are meant to indicate some complicated orbifold regions.
If S is an orientable embedded 2-orbifold in O then S is compressible if there is an embedded discal 2-orbifold D ⊂ O so that ∂D lies in S, but ∂D does not bound a discal 2-orbifold in S. (We call D a compressing discal orbifold.) Otherwise, S is incompressible.  Note that any embedded copy of a turnover S 2 (p, q, r) is automatically incompressible, since any embedded circle in S 2 (p, q, r) bounds a discal 2-orbifold in S 2 (p, q, r).
If O is a compact orientable 3-orbifold then there is a compact orientable irreducible 3orbifold O ′ so that O is the result of performing 0-surgeries on O ′ ; see [5,Chapter 3]. The orbifold O ′ can be obtained by taking an appropriate spherical system on O, cutting along the spherical 2-orbifolds and adding discal 3-orbifolds to the ensuing boundary components. If we take a minimal such spherical system then O ′ is canonical.
This shows that if O is a 3-manifold then O ′ is not just the disjoint components in the prime decomposition. That is, we are not dealing with a direct generalization of the Kneser-Milnor prime decomposition from 3-manifold theory. Because the notion of connected sum is more involved for orbifolds than for manifolds, the  . An essential spherical suborbifold notion of a prime decomposition is also more involved; see [51]. It is not needed for the present paper.
We assume now that O is irreducible. A Seifert 3-orbifold O fibers over a 2-dimensional orbifold, with circle fiber. We will only consider orientable Seifert 3-orbifolds.
As ∂O is an orientable 2-orbifold which fibers over a 1-dimensional orbifold, with circle fibers, any connected component of ∂O must be T 2 or S 2 (2, 2, 2, 2). In the case of a boundary component S 2 (2, 2, 2, 2), the generic fiber is a circle on |S 2 (2, 2, 2, 2)| which separates it into two 2-disks, each containing two singular points. That is, the pillowcase is divided into two half-pillowcases.
A solid-toric orbifold S 1 × D 2 or S 1 × D 2 (k) has an obvious Seifert fibering over D 2 or 2.5. Riemannian geometry of orbifolds. Definition 2.9. A Riemannian metric on an orbifold O is given by a collection of Riemannian metrics on the U α 's so that • G α acts isometrically on U α and • The embeddings ( U 3 , G 3 ) → ( U 1 , G 1 ) and ( U 3 , G 3 ) → ( U 2 , G 2 ) from part 5 of Definition 2.1 are isometric.
We say that the Riemannian orbifold O has sectional curvature bounded below by K ∈ R if the Riemannian metric on each U α has sectional curvature bounded below by K, and similarly for other curvature bounds.
A Riemannian orbifold has an orthonormal frame bundle F O, a smooth manifold with a locally free (left) O(n)-action whose quotient space is homeomorphic to |O|. Local charts for There is a horizontal distribution T H F O on F O coming from the Levi-Civita connection on U . If γ is a loop at p ∈ |O| then a horizontal lift of γ allows one to define the holonomy H γ , a linear map from T p O to itself.
If γ : [a, b] → O is a smooth map to a Riemannian orbifold then its length is L(γ) = b a |γ ′ (t)| dt, where |γ ′ (t)| can be defined by a local lifting of γ to a local model. This induces a length structure on |O|. The diameter of O is the diameter of |O|. We say that O is complete if |O| is a complete metric space. If O has sectional curvature bounded below by K ∈ R then |O| has Alexandrov curvature bounded below by K, as can be seen from the fact that the Alexandrov condition is preserved upon quotienting by a finite group acting isometrically [11,Proposition 10.2.4].
It is useful to think of O as consisting of an Alexandrov space equipped with an additional structure that allows one to make sense of smooth functions.
We write dvol for the n-dimensional Hausdorff measure on |O|. Using the above-mentioned relationship between the sectional curvature of O and the Alexandrov curvature of |O|, we can use [11, Chapter 10.6.2] to extend the Bishop-Gromov inequality from Riemannian manifolds with a lower sectional curvature bound, to Riemannian orbifolds with a lower sectional curvature bound. We remark that a Bishop-Gromov inequality for an orbifold with a lower Ricci curvature bound appears in [7].
A geodesic is a smooth curve γ which, in local charts, satisfies the geodesic equation. Any length-minimizing curve γ between two points is a geodesic, as can be seen by looking in a local model around γ(t).
Lemma 2.11. If O is a complete Riemannian orbifold then for any p ∈ |O| and any v ∈ T p O, there is a unique geodesic γ : R → O such that γ(0) = p and γ ′ (0) = v.
Proof. The proof is similar to the proof of the corresponding part of the Hopf-Rinow theorem, as in [37,Theorem 4.1].
The exponential map of a complete orbifold O is defined as follows. Given p ∈ |O| and v ∈ C p |O|, let γ : [0, 1] → O be the unique geodesic with |γ|(0) = p and |γ ′ |(0) = v. Put | exp |(p, v) = (p, |γ|(1)) ∈ |O| × |O|. This has the local lifting property to define a smooth orbifold map exp : Given p ∈ |O|, the restriction of exp to T p O gives an orbifold map exp p :  : f −1 (U) → U to U , we obtain a G p -equivariant Riemannian submersion f to U. If p ∈ U covers p then f −1 ( p) is a compact orbifold on which G p acts. Using the submersion structure, its normal bundle Theorem 9.42]. This passes to a diffeomorphism between f −1 (U) and ( f −1 ( p) × U )//G p .
If f : O 1 → O 2 is a local diffeomorphism and g 2 is a Riemannian metric on O 2 then there is a pullback Riemannian metric f * g 2 on O 1 , which makes f into a local isometry.
We now give a useful criterion for a local isometry to be a covering map.
Proof. The proof is along the lines of the corresponding manifold statement, as in [37,Theorem 4.6].
There is an orbifold version of the de Rham decomposition theorem. As in [37, Lemma IV. 6.6], one shows that (γ 1 (b), γ 2 (b)) only depends on γ(b). In this way, one defines a map f : O → O 1 × O 2 . As in [37, p. 192], one shows that f is a local isometry. As in [37, p. 188], one shows that O 1 and O 2 are simply-connected. The lemma now follows from Lemma 2.13.
The regular part |O| reg inherits a Riemannian metric. The corresponding volume form equals the n-dimensional Hausdorff measure on |O| reg . We define vol(O), or vol(|O|), to be the volume of the Riemannian manifold |O| reg , which equals the n-dimensional Hausdorff mass of the metric space |O|.
If f : O 1 → O 2 is a diffeomorphism between Riemannian orbifolds then we can define the C K -distance between g 1 and f * g 2 , using local models for O 1 . . Definition 2.15. Let (O 1 , p 1 ) and (O 2 , p 2 ) be pointed connected orbifolds with complete Riemannian metrics g 1 and g 2 that are C K -smooth. (That is, the orbifold transition maps are C K+1 and the metric tensor in a local model is C K .) Given ǫ > 0, we say that the C K -distance between (O 1 , p 1 ) and (O 2 , p 2 ) is bounded above by ǫ if there is a C K+1 -smooth map f :B(p 1 , ǫ −1 ) → O 2 that is a diffeomorphism onto its image, such that • The C K -distance between g 1 and f * g 2 on B(p 1 , ǫ −1 ) is at most ǫ, and Taking the infimum of all such possible ǫ's defines the C K -distance between (O 1 , p 1 ) and (O 2 , p 2 ).
Remark 2.16. It may seem more natural to require |f | to be basepoint-preserving. However, this would cause problems. For example, given k ≥ 2, take O = R 2 //Z k . Let π : R 2 → |O| be the quotient map. We would like to say that if i is large then the pointed orbifold (O, π(i −1 , 0)) is close to (O, π(0, 0)). However, there is no basepoint-preserving map f :B(π(i −1 , 0), 1) → (O, π(0, 0)) which is a diffeomorphism onto its image, due to the difference between the local groups at the two basepoints.
2.6. Critical point theory for distance functions. Let O be a complete Riemannian orbifold and let Y be a closed subset of |O|. A point p ∈ |O| − Y is noncritical if there is a nonzero G p -invariant vector v ∈ T p O ∼ = T p U making an angle strictly larger than π 2 with any lift to T p U of the initial velocity of any minimizing geodesic segment from p to Y .
In the next lemma we give an equivalent formulation in terms of noncriticality on |O|.
Lemma 2.17. A point p ∈ |O| − Y is noncritical if and only if there is some w ∈ C p |O| ∼ = T p U /G p so that the comparison angle between w and any minimizing geodesic from p to Y is strictly greater than π 2 .
Proof. Suppose that p is noncritical. Given v as in the definition of noncriticality, put w = vG p .
Conversely, suppose that w ∈ C p |O| ∼ = T p U /G p is such that the comparison angle between w and any minimizing geodesic from p to Y is strictly greater than π 2 . Let v 0 be a preimage of w in T p U. Then v 0 makes an angle greater than π 2 with the initial velocity of any minimizing geodesic from p to Y . As the set of such initial velocities is G p -invariant, for any g ∈ G p the vector v 0 g also makes an angle greater than π 2 with any lift to T p U of the initial velocity of any minimizing geodesic from p to Y . As {v 0 g} g∈Gp lies in an open half-plane, we can take v to be the nonzero vector 1 |Gp| g∈Gp v 0 g.
We now prove the main topological implications of noncriticality. Proof. The proof is similar to that of [12,Lemma 1.4]. For any p ∈ |O| − Y , there are a precompact neighborhood U p of p in |O| − Y and a smooth vector field V p on U p so that d Y has positive directional derivative in the V p direction, on U p . Let {U p i } be a finite collection that covers d −1 Y (a, b). From Lemma 2.6, there is a subordinate partition of unity {ρ i }. Put is a Lipschitz-regular suborbifold of O which is transversal to ξ, as can be seen in local models. Working in local models, inductively from lower-dimensional strata of |O| to higher-dimensional strata, we can slightly smooth O to form a smooth suborbifold O ′ of O which is transverse to ξ. Flowing (which is defined using local models) in the direction of ξ gives an orbifold diffeomorphism between O and R × O ′ .

Smoothing functions.
Let O be a Riemannian orbifold. Let F be a Lipschitz function on |O|. Given p ∈ |O|, we define the generalized gradient ∇ gen p F ⊂ T p O as follows. Let ( U , G) be a local model around p. Let F be the lift of F to U . Choose p ∈ U covering p. Let ǫ > 0 be small enough so that exp p : B(0, ǫ) → U is a diffeomorphism onto its image. If x ∈ B( p, ǫ) is a point of differentiability of F then compute ∇ x F and parallel transport it along the minimizing geodesic to p. Take the closed convex hull of the vectors so obtained and then take the intersection as ǫ → 0. This gives a closed convex G p -invariant subset of T p U , or equivalently a closed convex subset of T p O; we denote this set by ∇ gen p F . The union p∈|O| ∇ gen p F ⊂ T O will be denoted ∇ gen F . and for each p ∈ |O|, the preimage of (S ∩ |π| −1 (p)) ⊂ C p |O| in T p O is convex.) Then for any ǫ > 0 and any Lipschitz function F : |O| → R whose generalized gradient over U lies in S, there is a Lipschitz function F ′ : |O| → R such that : (1) There is an open subset of |O| containing C on which F ′ is a smooth orbifold function.
(2) The generalized gradient of F ′ , over U, lies in S.
Proof. The proof proceeds by mollifying the Lipschitz function F as in [26,Section 2]. The mollification there is clearly G-equivariant in a local model ( U, G). Let O be a complete Riemannian orbifold, let Y ⊂ |O| be a closed subset and let d Y : |O| → R be the distance function from Y . Given p ∈ |O| − Y , let V p ⊂ C p |O| be the set of initial velocities of minimizing geodesics from p to Y . Suppose that U ⊂ |O| − Y is an open subset such that for all p ∈ U, one has diam(V p ) < θ. Let C be a compact subset of U. Then for every ǫ 1 > 0, there is a Lipschitz function F ′ : |O| → R such that

Noncompact nonnegatively curved orbifolds
In this section we extend the splitting theorem and the soul theorem from Riemannian manifolds to Riemannian orbifolds. We give an argument to rule out tight necks in a noncompact nonnegatively curved orbifold. We give the topological description of noncompact nonnegatively curved orbifolds of dimension two and three. We may emphasize in some places that O is nonnegatively curved. Proof. As |O| contains a line, the splitting theorem for nonnegatively curved Alexandrov spaces [11,Chapter 10.5] implies that |O| is an isometric product R × Y for some complete nonnegatively curved Alexandrov space Y . The isometric splitting lifts to local models, Remark 3.4. A splitting theorem for orbifolds with nonnegative Ricci curvature appears in [8]. As the present paper deals with lower sectional curvature bounds, the more elementary Proposition 3.2 is sufficient for our purposes.
3.2. Cheeger-Gromoll-type theorem. A subset Z ⊂ |O| is totally convex if any geodesic segment (possibly not minimizing) with endpoints in Z lies entirely in Z.
Lemma 3.5. Let Z ⊂ |O| be totally convex and let ( U, G) be a local model. Put U = U/G and let q : U → U be the quotient map. If γ is a geodesic segment in U with endpoints in This is a contradiction.
the initial velocity of any minimizing geodesic from p to Y makes an angle greater than π 2 with v.
Proof. Using Lemma 3.5, the proof is along the lines of that in [25, Chapter 3.1].
We put ∂Z = Z − |N |. Note that in the definition of N we are dealing with orbifolds as opposed to manifolds.
Lemma 3.8. It is equivalent to require (3.7) for all geodesic segments or just for minimizing geodesic segments.
Proof. Suppose that (3.7) holds for all minimizing geodesic segments. Let γ : [a, b] → O be a geodesic segment, maybe not minimizing. For any t ∈ [a, b], we can find a neighborhood I t of t in [a, b] so that the restriction of γ to I t is minimizing. Then (3.7) holds on I t . It follows that (3.7) holds on [a, b].
Any superlevel set f −1 [c, ∞) of a concave function is closed and totally convex.
Let f be a proper concave function on |O| which is bounded above. Then there is a maximal c ∈ R so that the superlevel set Suppose for the rest of this subsection that O is noncompact. Fix a basepoint ⋆ ∈ |O|. Let η be a unit-speed ray in |O| starting from ⋆; note that η is automatically a geodesic. Let b η : |O| → R be the Busemann function; We now construct the soul of O, following Cheeger-Gromoll [14]. Let C 0 be the minimal nonempty superlevel set of f . For i ≥ 0, let C i+1 be the minimal nonempty superlevel set of Then S is a totally geodesic suborbifold of O.
Proof. Following [25, Lemma 3.3.1], we claim that d S has no critical points on |O| − S. To see this, choose p ∈ |O| − S. There is a totally convex set Z ⊂ |O| for which p ∈ ∂Z; either a superlevel set of f or one of the sets C i . Defining N as in Lemma 3.6, we also know that S ⊂ |N |. By Lemma 3.6, p is noncritical for d S .
. However, if ǫ is small then the normal exponential map gives a diffeomorphism between N S and O Nǫ(S) . Remark 3.14. One can define a soul for a general complete nonnegatively curved Alexandrov space X. The soul will be homotopy equivalent to X. However, X need not be homeomorphic to a fiber bundle over the soul, as shown by an example of Perelman [11, Example 10.10.9].
We include a result that we will need later about orbifolds with locally convex boundary.
Lemma 3.15. Let O be a compact connected orbifold-with-boundary with nonnegative sectional curvature. Suppose that ∂O is nonempty and has positive-definite second fundamental form. Then there is some p ∈ |O| so that ∂O is diffeomorphic to the unit distance sphere from the vertex in T p O.
Proof. Let p ∈ |O| be a point of maximal distance from |∂O|. We claim that p is unique. If not, let p ′ be another such point and let γ be a minimizing geodesic between them. Applying Lemma 3.9 with Z = |O|, there is a nontrivial geodesic s → V (s, d) of O that lies in |∂O|. This contradicts the assumption on ∂O. Thus p is unique. The lemma now follows from the proof of Lemma 3.13, as we are effectively in a situation where the soul is a point.

3.3.
Ruling out tight necks in nonnegatively curved orbifolds. Lemma 3.16. Suppose that O is a complete connected Riemannian orbifold with nonnegative sectional curvature. If X is a compact connected 2-sided codimension-1 suborbifold of O then precisely one of the following occurs : The remaining case is when X does not separate O. If γ is a smooth closed curve in O which is transversal to X (as defined in local models) then there is a well-defined intersection number γ · X ∈ Z. Since X is nonseparating, there is a γ so that γ · X = 0. We can use this notion of intersection number to define an infinite cyclic cover O ′ of O. As O ′ contains a line, the lemma follows from Proposition 3.2.
Proof. Let X be the preimage of X in R n . Let ∆ be any number greater than diam X (X). Let x be a point in |X|. Let { x i } i∈I be the preimages of x in X. Here the cardinality of I is bounded above by |G|. We claim that X = i∈I B( x i , ∆). To see this, let y be an arbitrary point in X. Let y be its image in X. Join y to x by a minimizing geodesic γ, which is necessarily of length at most ∆. Then a horizontal lift of γ, starting at y, joins y to some x i and also has length at most ∆.
Let C be a connected component of X. Since C is connected, it has a covering by a subset of {B( x i , 2 diam X (X))} i∈I with connected nerve, and so C has diameter at most 4|G| diam X (X). Furthermore, from the Jordan separation theorem, C is the boundary of a domain D ∈ R n with extrinsic diameter at most 4|G| diam X (X). Letting D ∈ O be the projection of D, the lemma follows.
Proposition 3.18. Suppose that O is a complete connected noncompact Riemannian norbifold, n > 2, with nonnegative sectional curvature. Then there is a number δ > 0 (depending on O) so that the following holds. Let X be a connected compact 2-sided codimension-1 suborbifold of O. Then either Proof. Suppose that the proposition is not true. Then there is a sequence {X i } ∞ i=1 of connected compact 2-sided codimension-1 suborbifolds of O so that lim i→∞ diam(X i ) = 0 but each X i fails to bound a connected suborbifold whose extrinsic diameter is at most 8 sup p∈|O| |G p | times as much.
If all of the |X i |'s lie in a compact subset of |O| then a subsequence converges in the Hausdorff topology to a point p ∈ |O|. As a sufficiently small neighborhood of p can be well approximated metrically by a neighborhood of 0 ∈ |R n //G p | after rescaling, Lemma 3.17 implies that for large i we can find . This is a contradiction. Hence we can assume that the sets |X i | tend to infinity.
If some X i does not bound a compact suborbifold of O then by Lemma 3.16, there is an If O had more than one end then it would split off an R-factor and as before, the sequence Hence O is one-ended and after passing to a subsequence, we can assume that D 1 ⊂ D 2 ⊂ . . .. Fix a basepoint ⋆ ∈ |D 1 |. Let η be a unit-speed ray in |O| starting from ⋆ and let b η be the Busemann function from (3.10).
. For t large, consider a geodesic triangle with vertices p, p ′ , η(t). Given X i with i large, if t is sufficiently large then pη(t) and p ′ η(t) pass through X i . Taking t → ∞, triangle comparison implies that d(p, p ′ ) ≤ diam(X i ). Taking i → ∞ gives p = p ′ . Thus b η is injective. This is a contradiction.

Nonnegatively curved 2-orbifolds.
Lemma 3.19. Let O be a complete connected orientable 2-dimensional orbifold with nonnegative sectional curvature which is C K -smooth, K ≥ 3. We have the following classification of the diffeomorphism type, based on the number of ends. For notation, Γ denotes a finite subgroup of the oriented isometry group of the relevant orbifold and Σ 2 denotes a simplyconnected bad 2-orbifold with some Riemannian metric.
• 0 ends : Proof. If O has zero ends then it is compact and the classification follows from the orbifold Gauss-Bonnet theorem [5, Proposition 2.9]. If O has more than one end then Proposition 3.2 implies that O has two ends and isometrically splits off an R-factor. Hence it must be diffeomorphic to R × S 1 . Suppose that O has one end. The soul S has dimension 0 or 1. If S has dimension zero then S is a point and O is diffeomorphic to the normal bundle of S, which is R 2 //Γ. If S has dimension one then it is S 1 or S 1 //Z 2 and O is diffeomorphic to the normal bundle of S. As S 1 × R has two ends, the only possibility is S 1 × Z 2 R.

Noncompact nonnegatively curved 3-orbifolds.
Lemma 3.20. Let O be a complete connected noncompact orientable 3-dimensional orbifold with nonnegative sectional curvature which is C K -smooth, K ≥ 3. We have the following classification of the diffeomorphism type, based on the number of ends. For notation, Γ denotes a finite subgroup of the oriented isometry group of the relevant orbifold and Σ 2 denotes a simply-connected bad 2-orbifold with some Riemannian metric.
Proof. Because O is noncompact, it has at least one end. If it has more than one end then Proposition 3.2 implies that O has two ends and isometrically splits off an R-factor. This gives rise to the possibilities listed for two ends.
Suppose that O has one end. The soul S has dimension 0, 1 or 2. If S has dimension zero then S is a point and O is diffeomorphic to the normal bundle of S, which is R 3 //Γ. If S has dimension one then it is 3.6. 2-dimensional nonnegatively curved orbifolds that are pointed Gromov-Hausdorff close to an interval. We include a result that we will need later about 2dimensional nonnegatively curved orbifolds that are pointed Gromov-Hausdorff close to an interval.
Lemma 3.21. There is some β > 0 so that the following holds. Suppose that O is a pointed nonnegatively curved complete orientable Riemannian 2-orbifold which is C K -smooth for some K ≥ 3. Let ⋆ ∈ |O| be a basepoint and suppose that the pointed ball (B(⋆, 10), ⋆) ⊂ |O| has pointed Gromov-Hausdorff distance at most β from the pointed interval ([0, 10], 0). Then for every r ∈ [1,9], the orbifold O is a discal 2-orbifold or is diffeomorphic to D 2 (2, 2). is not a discal 2-orbifold then it has at least two singular points, say p 1 , p 2 ∈ |O|. Choose q ∈ |O| with d(⋆, q) = 2. By triangle comparison, the comparison angles satisfy ∠ p 1 (p 2 , q) ≤ 2π Suppose that there are three distinct singular points p 1 , p 2 , p 3 ∈ |O|. We know that they lie in B(⋆, 1). Let p i q and p k p j denote minimal geodesics. If β is small then the angle at p 1 between p 1 q and p 1 p 2 is close to π 2 , and similarly for the angle at p 1 between p 1 q and p 1 p 3 . As dim(O) = 2, and p 1 has total cone angle π, it follows that if β is small then the angle at p 1 between p 1 p 2 and p 1 p 2 is small. The same reasoning applies at p 2 and p 3 , so we have a geodesic triangle in |O| with small total interior angle, which violates the fact that |O| has nonnegative Alexandrov curvature.

Riemannian compactness theorem for orbifolds
In this section we prove a compactness result for Riemannian orbifolds.
The statement of the compactness result is slightly different from the usual statement for Riemannian manifolds, which involves a lower injectivity radius bound. The injectivity radius is not a useful notion for orbifolds. For example, if O is an orientable 2-orbifold with a singular point p then a geodesic from a regular point q in |O| to p cannot minimize beyond p. As q could be arbitrarily close to p, we conclude that in any reasonable sense the injectivity radius of O would vanish.
Instead, our compactness result is phrased in terms of local volumes. This fits well with Perelman's work on Ricci flow, where local volume estimates arise naturally.
If one tried to prove a compactness result for Riemannian orbifolds directly, following the proofs in the case of Riemannian manifolds, then one would have to show that orbifold singularities do not coalesce when taking limits. We avoid this issue by passing to orbifold frame bundles, which are manifolds, and using equivariant compactness results there.
Compactness theorems for Riemannian metrics and Ricci flows for orbifolds with isolated singularities were proved in [38]. Compactness results for general orbifolds were stated in [16,Chapter 3.3] with a short sketch of a proof. [24,Section 6], after taking a subsequence we may assume that the frame bundles Gromov-Hausdorff topology to a C K−1 -smooth Riemannian manifold X with an isometric O(n)action and a basepoint p ∞ . (We lose one derivative because we are working on the frame bundle.) Furthermore, we may assume that the convergence is realized as follows : Given any O(n)-invariant compact codimension-zero submanifold-with-boundary K ⊂ X, for large i there is an O(n)-invariant compact codimension-zero submanifold-with-boundary K i ⊂ F O i and a smooth O(n)-equivariant fiber bundle K i → K with nilmanifold fiber whose diameter goes to zero as i → ∞ [13, Section 3], [24, Section 9].
Quotienting by O(n), the underlying spaces [11,Corollary 10.10.11]. Thus there is no collapsing and so for large i the submersion In particular, the O(n)-action on X is locally free. There is a corresponding quotient orbifold That is, under the hypotheses of Proposition 4.1, the orders of the isotropy groups cannot increase in the limit.
Remark 4.3. In the proof of Proposition 4.1, the submersions K i → K may not be basepointpreserving. This is where one has to leave the world of basepoint-preserving maps.

Ricci flow on orbifolds
In this section we first make some preliminary remarks about Ricci flow on orbifolds and we give the orbifold version of Hamilton's compactness theorem. We then give the topological classification of compact nonnegatively curved 3-orbifolds. Finally, we extend Perelman's no local collapsing theorem to orbifolds. 5.1. Function spaces on orbifolds. Let ρ : O(n) → R N be a representation. Given a local model ( U α , G α ) and a G α -invariant Riemannian metric on U α , let V α = R N × O(n) F U α be the associated vector bundle. If O is a n-dimensional Riemannian orbifold then there is an associated orbivector bundle V with local models ( V α , G α ). Its underlying space is |V | = R N × O(n) F O. By construction, V has an inner product coming from the standard inner product on R N . A section s of V is given by an O(n)-equivariant map s : F O → R N . In terms of local models, s is described by G α -invariant sections s α of V α that satisfy compatibility conditions with respect to part 5 of Definition 2.1.
The C K -norm of s is defined to be the supremum of the C K -norms of the s α 's. Similarly, the square of the H K -norm of s is defined to be the integral over |O| reg of the local square H K -norm, the latter being defined using local models. (Note that |O| reg has full Hausdorff n-measure in |O|.) Then H −K can be defined by duality. One has the rough Laplacian One can define differential operators and pseudodifferential operators acting on H Ksections of V . Standard elliptic and parabolic regularity theory extends to the orbifold setting, as can be seen by working equivariantly in local models.

5.2.
Short-time existence for Ricci flow on orbifolds. Suppose that {g(t)} t∈[A,B] is a smooth 1-parameter family of Riemannian metrics on O. We will call g a flow of metrics on O. The Ricci flow equation ∂g ∂t = − 2 Ric makes sense in terms of local models. Using the deTurck trick [18], which is based on local differential analysis, one can reduce the shorttime existence problem for the Ricci flow to the short-time existence problem for a parabolic PDE. Then any short-time existence proof for parabolic PDEs on compact manifolds, such as that of [54,Proposition 15.8.2], will extend from the manifold setting to the orbifold setting.
Remark 5.1. Even in the manifold case, one needs a slight additional argument to reduce the the short-time existence of the Ricci-de Turck equation to that of a standard quasilinear parabolic PDE. In local coordinates the Ricci-de Turck equation takes the form There is a slight issue since (5.2) is not uniformly parabolic, in that g kl could degenerate with respect to, say, the initial metric g 0 . This issue does not seem to have been addressed in the literature. However, it is easily circumvented. Let M be the space of smooth Riemannian metrics on M. Let F : M → M be a smooth map so that for some ǫ > 0, we have F (g) = g if g − g 0 g 0 < ǫ, and in addition ǫg 0 ≤ F (g) ≤ ǫ −1 g 0 for all g. (Such a map F is easily constructed using the fact that the inner products on T p M, relative to g 0 (p), can be identified with GL(n, R)/O(n), along with the fact that GL(n, R)/O(n) deformation retracts onto a small ball around its basepoint.) By [54,Proposition 15.8.2], there is a short-time solution to with g(0) = g 0 . Given this solution, there is some δ > 0 so that g(t) − g 0 g 0 < ǫ whenever t ∈ [0, δ]. Then {g(t)} t∈[0,δ] also solves the Ricci-de Turck equation (5.2).
We remark that any Ricci flow results based on the maximum principle will have evident extensions from manifolds to orbifolds. Such results include • The lower bound on scalar curvature • The Hamilton-Ivey pinching results for three-dimensional scalar curvature • Hamilton's differential Harnack inequality for Ricci flow solutions with nonnegative curvature operator • Perelman's differential Harnack inequality.

5.3.
Ricci flow compactness theorem for orbifolds. Let O 1 and O 2 be two connected pointed n-dimensional orbifolds, with flows of metrics g 1 and g 2 . If f : O 1 → O 2 is a (time-independent) diffeomorphism then we can construct the pullback flow f * g 2 and define the C K -distance between g 1 and f * g 2 , using local models for O 1 . . Suppose that g i (t) is complete for each t. Given ǫ > 0, suppose that f :B(p 1 , ǫ −1 ) → O 2 is a smooth map from the time-zero ball that is a diffeomorphism onto its image. Let |f | : B(p 1 , ǫ −1 ) → |O 2 | be the underlying map. We say that the C K -distance between the flows (O 1 , Taking the infimum of all such possible ǫ's defines the C K -distance between the flows (O 1 , p 1 , g 1 ) and (O 2 , p 2 , g 2 ).
Note that time derivatives appear in the definition of the C K -distance between g 1 and f * g 2 .
Suppose that the following two conditions are satisfied : 1. There is a function N : Then a subsequence of the solutions converges to a Ricci flow solution Proof. Using Proposition 4.1, the proof is essentially the same as that in [29, p. 548-551] and [38, p. 1116-1117] 5.4. Compact nonnegatively curved 3-orbifolds.
Proposition 5.6. Any compact nonnegatively curved 3-orbifold O is diffeomorphic to one of a simply-connected bad 2-orbifold equipped with its unique Ricci soliton metric [58, Proof. Let k be the largest number so that the universal cover O isometrically splits off an If O ′ is noncompact then by the Cheeger-Gromoll argument [15, Pf. of Theorem 3], |O ′ | contains a line. Proposition 3.2 implies that O ′ splits off an R-factor, which is a contradiction. Thus O ′ is simply-connected and compact with nonnegative sectional curvature.
If k = 2 then there is a contradiction, as there is no simply-connected compact 1-orbifold.
If k = 1 then O ′ is diffeomorphic to S 2 or Σ 2 . The Ricci flow on O = R × O ′ splits isometrically. After rescaling, the Ricci flow on O ′ converges to a constant curvature metric on S 2 or to the unique Ricci soliton metric on Σ 2 [58]. Hence π 1 (O) is a subgroup of Isom(R×S 2 ) or Isom(R×Σ 2 ), where the isometry groups are in terms of standard metrics. As π 1 (O) acts properly discontinuously and cocompactly on O, there is a short exact sequence where Γ 1 ⊂ Isom(O ′ ) and Γ 2 is an infinite cyclic group or an infinite dihedral group. It follows that O is finitely covered by S 1 × S 2 or S 1 × Σ 2 .
Suppose that k = 0. If O is positively curved then any proof of Hamilton's theorem about 3-manifolds with positive Ricci curvature [30] extends to the orbifold case, to show that O admits a metric of constant positive curvature; c.f. [33]. Hence we can reduce to the case when O does not have positive curvature and the Ricci flow does not immediately give it positive curvature. From the strong maximum principle as in [28,Section 8], for any p ∈ |O| reg there is a nontrivial orthogonal splitting T p O = E 1 ⊕ E 2 which is invariant under holonomy around loops based at p. The same will be true on O. Lemma 2.14 implies that O splits off an R-factor, which is a contradiction.
where the scalar curvature R and the normγ(τ )| are evaluated using the metric at time Given an L-geodesic γ, its initial velocity is defined to be where the infimum runs over piecewise smooth curves γ with |γ|(0) = p and |γ|(τ ) = q. Then any piecewise smooth curve γ which is a minimizer for L is a smooth L-geodesic.
Lemma 5.11. There is a minimizer γ for L.
Proof. The proof is similar to that in [35, p. 2631]. We outline the steps. Given p and q, one considers piecewise smooth curves γ as above. Fixing ǫ > 0, one shows that the curves γ with L(γ) < L(q, τ ) + ǫ are uniformly continuous. In particular, there is an R < ∞ so that any such γ lies in B(p, R). Next, one shows that there is some ρ ∈ (0, R) so that for any x ∈ B(p, R), there is a local model ( U , G x ) with U/G x = B(x, ρ) such that for any p ′ , q ′ ∈ B(x, ρ) and any subinterval [τ 1 , • There is a unique minimizer for the functional minimizing γ is smooth and its image lies in B(x, ρ). This is shown by working in the local models. Now cover B(p, R) by a finite number of ρ-balls Using the uniform continuity, let A ∈ Z + be such that for any γ : [0, τ ] → O with γ(0) = p, γ(τ ) = q and L(γ) < L(q, τ ) + ǫ, and any [τ 1 , τ 2 ] ⊂ [0, τ ] of length at most τ A , the distance between γ(τ 1 ) and γ(τ 2 ) is less than the Lebesgue number of the covering. We can effectively reduce the problem of finding a minimizer for L to the problem of minimizing a continuous function defined on tuples (p 0 , . . . , p A ) ∈ B(p, R) A+1 with p 0 = p and p A = q.
This shows that the minimizer exists.
Define the L-exponential map : T p O → O by saying that for v ∈ C p |O|, we put L exp τ (v) = γ(τ ), where γ is the unique L-geodesic from p whose initial velocity is v. Then L exp τ is a smooth orbifold map. Proof. The proof is similar to that in [35, p. 2632]. By Sard's theorem, it suffices to show that the subset B ′ τ ⊂ B τ , consisting of regular values of L exp τ , has measure zero in |O|. One shows that B ′ τ is contained in the underlying spaces of a countable union of codimension-1 suborbifolds of O, which implies the lemma. Therefore one may compute the integral of any integrable function on |O| by pulling it back to Ω τ ⊂ C p |O| and using the change of variable formula.
Proof. The proof is similar to that in [35,Section 23]. In the proof, one pulls back the integrand to C p |O|.
Proof. The proof is similar to that in [35,Section 24]. It uses the maximum principle, which is valid for orbifolds.
Lemma 5.17. If a Ricci flow solution is κ-noncollapsed on some scale then there is a uniform upper bound |G p | ≤ N(n, κ) on the orders of the isotropy groups at points p ∈ |O|.
Proof. The proof is similar to that in [35, Section 28].

κ-solutions
In this section we extend results about κ-solutions from manifolds to orbifolds. 6.1. Asymptotic solitons. Let (p, t 0 ) be a point in a κ-solution (O, g(·)) so that G p has maximal order. Define the reduced volume V (τ ) and the reduced length l(q, τ ) as in Subsection 5.5, by means of curves starting from (p, t 0 ), with τ = t 0 − t. From Lemma 5.15, for each τ > 0 there is some q(τ ) ∈ |O| such that l(q(τ ), τ ) ≤ n 2 . (Note that l ≥ 0 from the curvature assumption.) Proof. The proof is similar to that in [35,Section 39]. Using estimates on the reduced length as defined with the basepoint (p, t 0 ), one constructs a limit Ricci flow solution (O ∞ , g ∞ (·)) defined for t ∈ [−1, − 1 2 ], which is a gradient shrinking soliton. The only new issue is to show that it is nonflat.
As in [35,Section 39], there is a limiting reduced length function l ∞ (·, τ ) ∈ C ∞ (O), and a reduced volume which is a constant c, strictly less than the t → t 0 limit of the reduced volume of (O, g(·)). The latter equals (4π) n 2 |Gp| . If the limit solution were flat then l ∞ (·, τ ) would have a constant positive-definite Hessian. It would then have a unique critical point q. Using the gradient flow of l ∞ (·, τ ), one deduces that O ∞ is diffeomorphic to T q O ∞ . As in [35,Section 39], one concludes that As |G q | ≤ |G p |, we obtain a contradiction.
Lemma 6.4. Any two-dimensional κ-solution (O, g(·)) is an isometric quotient of the round shrinking 2-sphere or is a Ricci soliton metric on a bad 2-orbifold.
Proof. The proof is similar to that in [60, Lemma 3]. One considers the asymptotic soliton and shows that it has strictly positive scalar curvature outside of a compact region (as in [48,Lemma 1.2]). Using standard Jacobi field estimates, the asymptotic soliton must be compact. The lemma then follows from convergence results for 2-dimensional compact Ricci flow (using [58] in the case of bad 2-orbifolds).
Remark 6.5. One can alternatively prove Lemma 6.4 using the fact that if (O, g(·)) is a κ-solution then so is the pullback solution ( O, g(·)) on the universal cover. If O is a bad 2-orbifold then O is compact and the result follows from [58]. If O is a good 2-orbifold then ( O, g(·)) is a round shrinking S 2 from [35, Section 40].
6.3. Asymptotic scalar curvature and asymptotic volume ratio. Definition 6.6. If O is a complete connected Riemannian orbifold then its asymptotic scalar curvature ratio is R = lim sup q→∞ R(q)d(x, p) 2 . It is independent of the basepoint p ∈ |O|. Suppose first that n ≥ 3. By Lemma 3.15, for large k there is some p k ∈ |O| so that the suborbifold S k is diffeomorphic to the unit sphere in T p k O. As S k is diffeomorphic to S ∞ for large k, we conclude that S ∞ is isometric to S n−1 //Γ for some finite group Γ ⊂ Isom + (S n−1 ). Let p ∈ |O| be a point with G p ∼ = Γ. As C T |O| is isometric to R n /Γ, lim r→∞ r −n vol(B(p, r)) exists and equals the 1 |Γ| times the volume of the unit ball in R n . On the other hand, this equals lim r→0 r −n vol(B(p, r)). As we have equality in the Bishop-Gromov inequality, we conclude that O is flat, which is a contradiction.
If n = 2 then we can adapt the argument in [35,Section 41] to the orbifold setting. Definition 6.8. If O is a complete n-dimensional Riemannian orbifold with nonnegative Ricci curvature then its asymptotic volume ratio is V = lim r→∞ r −n vol(B(p, r)). It is independent of the choice of basepoint p ∈ |O|. Lemma 6.9. Let (O, g(·)) be a noncompact κ-solution. Then the asymptotic volume ratio V vanishes for each time slice (O, g(t 0 )). Moreover, there is a sequence of points p k ∈ |O| going to infinity such that the pointed sequence {(O, (p k , t 0 ), g(·)} ∞ k=1 converges, modulo rescaling by R(p k , t 0 ), to a κ-solution which isometrically splits off an R-factor.
Proof. The proof is similar to that in [35,Section 41] 6.4. In a κ-solution, the curvature and the normalized volume control each other. Lemma 6.10. Given n ∈ Z + , we consider n-dimensional κ-solutions.
(1) If B(p 0 , r 0 ) is a ball in a time slice of a κ-solution then the normalized volume r −n vol(B(p 0 , r 0 )) is controlled (i.e. bounded away from zero) ⇐⇒ the normalized scalar curvature r 2 0 R(p 0 ) is controlled (i.e. bounded above) k=1 is a sequence of pointed κ-solutions and for some r > 0, the r-balls B(p k , r) ⊂ (O k , g k (t k )) have controlled normalized volume, then a subsequence converges to an ancient solution (O ∞ , (p ∞ , 0), g ∞ (·)) which has nonnegative curvature operator, and is κ-noncollapsed (though a priori the curvature may be unbounded on a given time slice).
(3) There is a constant η = η(n, κ) so that for all p ∈ |O|, we have |∇R|(p, t) ≤ ηR 3 2 (p, t) and |R t |(p, t) ≤ ηR 2 (p, t). More generally, there are scale invariant bounds on all derivatives of the curvature tensor, that only depend on n and κ.
Proof. The proof is similar to that in [35,Section 42]. In the proof by contradiction of the implication ⇐= of part (1), after passing to a subsequence we can assume that |G p k | is a constant C. Then we use the argument in [35,Section 42] with c n equal to 1 C times the volume of the unit Euclidean n-ball.
Proof. The proof is similar to that in [35, Section 45].

6.7.
Compactness of the space of three-dimensional κ-solutions.
Proof. If {(O k , (p k , 0), g k (·))} ∞ k=1 is a sequence of such κ-solutions with R(p k , 0) = 1 then parts (1) and (2) of Lemma 6.10 imply that there is a subsequence that converges to an ancient solution (O ∞ , (p ∞ , 0), g ∞ (·)) which has nonnegative curvature operator and is κnoncollapsed. The remaining issue is to show that it has bounded curvature. Since R t ≥ 0, it is enough to show that (O ∞ , g ∞ (0)) has bounded scalar curvature.
If not then there is a sequence of points q i going to infinity in |O ∞ | such that R(q i , 0) → ∞ and R(q, 0) ≤ 2R(q i , 0) for q ∈ B(q i , A i R(q i , 0) − 1 2 ), where A i → ∞. Using the κnoncollapsing, a subsequence of the rescalings (O ∞ , q i , R(q i , 0)g ∞ ) will converge to a limit orbifold N ∞ that isometrically splits off an R-factor. By Lemma 6.4, N ∞ must be a standard solution on R × (S 2 //Γ) or R × (Σ 2 //Γ). Thus (O ∞ , g ∞ ) contains a sequence X i of neck regions, with their cross-sectional radii tending to zero as i → ∞. This contradicts Proposition 3.18. 6.8. Necklike behavior at infinity of a three-dimensional κ-solution.
Definition 6.18. Fix ǫ > 0. Let (O, g(·)) be an oriented three-dimensional κ-solution. We say that a point p 0 ∈ |O| is the center of an ǫ-neck if the solution g(·) in the set 0), is, after scaling with the factor Q, ǫ-close in some fixed smooth topology to the corresponding subset of a κ-solution R × O ′ that splits off an R-factor. That is, O ′ is the standard evolving S 2 //Γ or Σ 2 //Γ with extinction time 1. Here Σ 2 is a simply-connected bad 2-orbifold with a Ricci soliton metric.
We let |O| ǫ denote the points in |O| which are not centers of ǫ-necks. Proposition 6.19. For all κ > 0, there exists an ǫ 0 > 0 such that for all 0 < ǫ < ǫ 0 there exists an α = α(ǫ, κ) with the property that for any oriented three dimensional κ-solution (O, g(·)), and at any time t, precisely one of the following holds : • (O, g(·)) splits off an R-factor and so every point at every time is the center of an ǫ-neck for all ǫ > 0.
• O is compact, and there is a pair of points x, y ∈ |O| ǫ such that R(x) d 2 (x, y) > α, and there is a minimizing geodesic xy such that every z ∈ |O| − |O| ǫ satisfies R(z) d 2 (z, xy) < α. • O is compact and there exists a point x ∈ |O| ǫ such that R(x) d 2 (x, z) < α for all z ∈ |O|.
Proof. The proof is similar to that in [35, Section 48].
Lemma 6.21. Any three-dimensional gradient shrinking κ-solution O is one of the following: • A finite isometric quotient of the round shrinking S 3 .  Lemma 6.23. Given N ∈ Z + , there is a κ 0 = κ 0 (N) > 0 so that if (O, g(·)) is an oriented three-dimensional κ-solution for some κ > 0, with |G p | ≤ N for all p ∈ |O|, then it is a κ 0 -solution or it is a quotient of the round shrinking S 3 .
Proof. The proof is similar to that in [35,Section 50]. The bound on |G p | gives a finite number of possible noncompact asymptotic solitons from Lemma 6.21, since a given closed two-dimensional orbifold has a unique Ricci soliton metric up to scaling, and the topological type of S 2 //Γ (or Σ//Γ) is determined by the number of singular points (which is at most three) and the isotropy groups of those points.. Lemma 6.24. Given N ∈ Z + , there is a universal constant η = η(N) > 0 such that at each point of every three-dimensional ancient solution (O, g(·)) that is a κ-solution for some κ > 0, and has |G p | ≤ N for all p ∈ |O|, we have estimates Proof. The proof is similar to that in [35, Section 59].

Ricci flow with surgery for orbifolds
In this section we construct the Ricci-flow-with-surgery for three-dimensional orbifolds.
Starting in Subsection 7.2, we will assume that there are no bad 2-dimensional suborbifolds. Starting in Subsection 7.5, we will assume that the Ricci flows have normalized initial conditions, as defined there. Rm(p, t) ≥ −Φ(R(p, t)).
Our example of Φ-almost nonnegative curvature comes from the Hamilton-Ivey pinching condition [35,Appendix B], which is valid for any three-dimensional orbifold Ricci flow solution which has complete time slices, bounded curvature on compact time intervals, and initial curvature operator bounded below by −I. Proposition 7.3. Given ǫ, κ, σ > 0 and a function Φ as above, one can find r 0 > 0 with the following property. Let (O, g(·)) be a Ricci flow solution on a three-dimensional orbifold O, defined for 0 ≤ t ≤ T with T ≥ 1. We suppose that for each t, g(t) is complete, and the sectional curvature in bounded on compact time intervals. Suppose that the Ricci flow has Φ-almost nonnegative curvature and is κ-noncollapsed on scales less than σ. Then for any point (p 0 , t 0 ) with t 0 ≥ 1 and Q = R(p 0 , t 0 ) ≥ r −2 0 , the solution in {(p, t) : dist 2 t 0 (p, p 0 ) < (ǫQ) −1 , t 0 − (ǫQ) −1 ≤ t ≤ t 0 } is, after scaling by the factor Q, ǫ-close to the corresponding subset of a κ-solution.
Proof. The proof is similar to that in [35,Section 52]. We have to allow for the possibility of neck-like regions approximated by R × (S 2 //Γ) or R × (Σ 2 //Γ). On the last two lines of [35, p. 2702], the "injectivity radius" can be replaced by the "local volume".

Necks and horns.
Assumption 7.4. Hereafter, we only consider three-dimensional orbifolds that do not contain embedded bad 2-dimensional suborbifolds.
In particular, neck regions will be modeled on R × (S 2 //Γ), where S 2 //Γ is a quotient of the round shrinking S 2 .
We let B(p, t, r) denote the open metric ball of radius r, with respect to the metric at time t, centered at p ∈ |O|.
We let P (p, t, r, ∆t) denote a parabolic neighborhood, that is the set of all points (p ′ , t ′ ) with p ′ ∈ B(p, t, r) and t ′ ∈ [t, t + ∆t] or t ′ ∈ [t + ∆t, t], depending on the sign of ∆t.  Definition 7.7. A metric on (−1, 1) × (S 2 //Γ) such that each point is contained in an ǫ-neck is called an ǫ-tube, an ǫ-horn or a double ǫ-horn if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, or tends to infinity on both ends, respectively.
A metric on B 3 //Γ or (−1, 1) × Z 2 (S 2 //Γ), such that each point outside some compact subset is contained in an ǫ-neck, is called an ǫ-cap or a capped ǫ-horn, if the scalar curvature stays bounded or tends to infinity on the end, respectively.  Lemma 7.11. If a three-dimensional κ-solution (O, g(·)) is compact and has a noncompact asymptotic soliton then O is diffeomorphic to S 3 //Z k or S 3 //D k for some k ≥ 1.
Proof. The proof is similar to that in [35, Section 59].
Lemma 7.12. For every sufficiently small ǫ > 0 one can find C 1 = C 1 (ǫ) and C 2 = C 2 (ǫ) such that for each point (p, t) in every κ-solution, there is a radius r ∈ [R(p, t) − 1 2 , C 1 R(p, t) − 1 2 ] and a neighborhood B, B(p, t, r) ⊂ B ⊂ B(p, t, 2r), which falls into one of the four categories : (a) B is a strong ǫ-neck, or (b) B is an ǫ-cap, or (c) B is a closed orbifold diffeomorphic to S 3 //Z k or S 3 //D k for some k ≥ 1. Furthermore: • The scalar curvature in B at time t is between C −1 2 R(p, t) and C 2 R(p, t). • The volume of B is cases (a), (b) and (c) is greater than C −1 2 R(p, t) − 3 2 . • In case (b), there is an ǫ-neck U ⊂ B with compact complement in B such that the distance from p to U is at least 10000R(p, t) − 1 2 . • In case (c) the sectional curvature in B is greater than C −1 2 R(p, t).
Proof. The proof is similar to that in [35, Section 59].

Standard solutions.
Put O = R 3 //Γ, where Γ is a finite subgroup of SO(3). We fix a smooth SO(3)-invariant metric g 0 on R 3 which is the result of gluing a hemispherical-type cap to a half-infinite cylinder [0, ∞) × S 2 of scalar curvature 1. We also use g 0 to denote the quotient metric on O. Among other properties, g 0 is complete and has nonnegative curvature operator. We also assume that g 0 has scalar curvature bounded below by 1. Lemma 7.14. Let (R 3 //Γ, g(·)) be a standard solution. Then: (1) The curvature operator of g is nonnegative.
(2) All derivatives of curvature are bounded for small time, independent of the standard solution.
(3) The blowup time is 1 and the infimal scalar curvature on the time-t slice tends to infinity as t → 1 − uniformly for all standard solutions.
(4) (R 3 //Γ, g(·)) is κ-noncollapsed at scales below 1 on any time interval contained in [0, 1), where κ depends only on g 0 and |Γ|. (5) (R 3 //Γ, g(·)) satisfies the conclusion of Proposition 7.3. Here v 0 is a global parameter in the sense that it will be fixed throughout the rest of the paper. If O is normalized then the Bishop-Gromov inequality implies that there is a uniform upper bound N = N(v 0 ) < ∞ on the order of the isotropy groups; cf. the proof of Lemma 5.17. The next lemma says that by rescaling we can always achieve a normalized metric.
Lemma 7.16. Given N ∈ Z + , there is a v 0 = v 0 (N) > 0 with the following property. Let O be a compact orientable Riemannian three-dimensional orbifold, whose isotropy groups have order at most N. Then a rescaling of O will have a normalized metric.
Proof. Let Ω 3 be the volume of the unit ball in R 3 . Bishop-Gromov implies that a ball B r of radius r > 0 with arbitrary center in a Euclidean orbifold R n //G, where G is a finite subgroup of O(3) with order at most N, has volume bounded below by Ω 3 N r 3 . Put v 0 = Ω 3 2N . We claim that this value of v 0 satisfies the lemma.
To prove this by contradiction, suppose that there is an orbifold O which satisfies the hypotheses of the lemma but for which the conclusion fails. Then there is a sequence After passing to a subsequence, we can assume that lim i→∞ p i = p ′ for some p ′ ∈ |O|. Using the inverse exponential map, for large i the ball B(p i , r i ) will, up to small distortion, correspond to a ball of radius r i in the tangent space T p ′ O. In view of our choice of v 0 , this is a contradiction.
Assumption 7.17. Hereafter we assume that our Ricci flows have normalized initial condition.
Consider the labels on the edges in the singular part of the orbifold. They clearly do not change under a smooth Ricci flow. If some components of the orbifold are discarded at a singularity time then the set of edge labels can only change by deletion of some labels. Otherwise, the surgery procedure will be such that the set of edge labels does not change, although the singular graphs will change. Hence the normalized initial condition implies a uniform upper bound on the orders of the isotropy groups for all time.
Let O be a connected closed oriented 3-dimensional orbifold. Let g(·) be a Ricci flow on O defined on a maximal time interval [0, T ) with T < ∞. For any ǫ > 0, we know that there are numbers r = r(ǫ) > 0 and κ = κ(ǫ) > 0 so that for any point (p, t) with Q = R(p, t) ≥ r −2 , the solution in P (p, t, (ǫQ) − 1 2 , (ǫQ) −1 ) is (after rescaling by the factor Q) ǫ-close to the corresponding subset of a κ-solution.
• Any connected component of Ω is noncompact.  The r-canonical neighborhood assumption is the obvious orbifold extension of what's in [35,Section 69], with condition (c) replaced by "O is a closed orbifold diffeomorphic to an isometric quotient of S 3 ".
The Φ-pinching assumption is the same as in [35,Section 69].
The a priori assumptions consist of the Φ-pinching assumption and the r-canonical neighborhood assumption.
Lemma 7.25. Given the pinching function Φ, a number T ∈ (0, ∞), a positive nonincreasing function r : [0, T ] → R and a number δ ∈ (0, 1 2 ), there is a nonincreasing function h : [0, T ] → R with 0 < h(t) < δ 2 r(t) so that the following property is satisfied. Let M be a Ricci flow with surgery defined on [0, T ), with T < T , which satisfies the a priori assumptions and which goes singular at time T . Let (Ω, g) denote the time-T limit. Put ρ = δr(T ) and Suppose that (p, T ) lies in an ǫ-horn H ⊂ Ω whose boundary is contained in Ω ρ . Suppose also that R(p, T ) ≥ h −2 (T ). Then the parabolic region P (p, Proof. The proof is the same as that in [35, Section 72], working equivariantly. We define a Ricci flow with (r, δ)-cutoff by the obvious orbifold extension of the definition in [35,Section 73].
In the surgery procedure, one first throws away all connected components of Ω which do not intersect Ω ρ . For each connected component Ω j of Ω that intersects Ω ρ and for each ǫ-horn of Ω j , take a cross-sectional S 2 -quotient that lies far in the ǫ-horn. Let X be what's left after cutting the ǫ-horns at the 2-sphere quotients and removing the tips. The (possibly disconnected) postsurgery orbifold O ′ is the result of capping off ∂X by discal 3-orbifolds.
Lemma 7.28. The presurgery orbifold can be obtained from the postsurgery orbifold by applying the following operations finitely many times: • Taking the disjoint union with a finite isometric quotient of S 1 × S 2 or S 3 . Remark 7.31. We restrict to 3-orbifolds without bad 2-suborbifolds in order to perform surgery. Without this assumption, there could be a neckpinch whose cross-section is a bad 2-orbifold Σ. In the case of a nondegenerate neckpinch, the blowup limit would be the product of R with an evolving Ricci soliton metric on Σ. The problem in performing surgery is that after slicing at a bad cross-section, there is no evident way to cap off the ensuing pieces with 3-dimensional orbifolds so as to preserve the Hamilton-Ivey pinching condition.

Hyperbolic regions
In this section we show that the w-thick part of the evolving orbifold approaches a finitevolume Riemannian orbifold with constant curvature − 1 4 . As a standing assumption in this section, we suppose that we have a solution to the Ricci flow with (r, δ)-cutoff and with normalized initial data.
(2) The ball B(p 0 , t 0 , r 0 ) has sectional curvatures at least −r −2 0 at each point. and sectional curvatures at least −r −2 t −1 0 then the curvature at (p 0 , t 0 ) satisfies (b) Given in addition A < ∞ and allowing T to depend on A, we can ensure (8.3) for all points in B(p 0 , t 0 , Ar √ t 0 ).

Proof
In what follows, we take "hyperbolic" to mean "constant curvature − 1 4 ". When applied to a hyperbolic orbifold, the definitions of the thick and thin parts are essentially equivalent to those in [5,Chapter 6.2], to which we refer for more information about hyperbolic 3orbifolds.
Recall that a hyperbolic 3-orbifold can be written as H 3 //Γ for some discrete group A rank-2 cusp neighborhood is the quotient of a horoball in H 3 by an elementary rank-2 parabolic group.
In either case, the boundary is a compact Euclidean 2-orbifold.
There is a Margulis constant µ 0 > 0 so that for any finite-volume hyperbolic 3-orbifold O, if µ ≤ µ 0 then the connected components of the µ-thin part of O are Margulis tubes or rank-2 cusp neighborhoods.
Furthermore, given a finite-volume hyperbolic 3-orbifold O, if µ > 0 is sufficiently small then the connected components of the µ-thin part are rank-2 cusp neighborhoods.
Mostow-Prasad rigidity works just as well for finite-volume hyperbolic orbifolds as for finite-volume hyperbolic manifolds. Indeed, the rigidity statements are statements about lattices in Isom(H n ).
defined for t ∈ [T 0 , ∞), such that (1) f (t) is close to an isometry: (2) f (t) defines a smooth family of maps which changes smoothly with time:

Locally collapsed 3-orbifolds
In this section we consider compact Riemannian 3-orbifolds O that are locally collapsed with respect to a local lower curvature bound. Under certain assumptions about smoothness and boundary behavior, we show that O is either the result of performing 0-surgery on a strong graph orbifold or is one of a few special types. We refer to Definition A.7 for the definition of a strong graph orbifold.
We first consider the boundaryless case.
Proposition 9.1. Let c 3 be the volume of the unit ball in R 3 , let K ≥ 10 be a fixed integer and let N be a positive integer. Fix a function A : (0, ∞) → (0, ∞). Then there is a w 0 ∈ (0, c 3 /N) such that the following holds.
Suppose that (O, g) is a connected closed orientable Riemannian 3-orbifold. Assume in addition that for all p ∈ |O|, holds in the ball B(p, r).
Then O is the result of performing 0-surgeries on a strong graph orbifold or is diffeomorphic to an isometric quotient of S 3 or T 3 . Remark 9.3. We recall that a strong graph orbifold is allowed to be disconnected. By Proposition A.10, a weak graph orbifold is the result of performing 0-surgeries on a strong graph orbifold. Because of this, to prove Proposition 9.1 it is enough to show that O is the result of performing 0-surgeries on a weak graph orbifold or is diffeomorphic to an isometric quotient of S 3 or T 3 .
Remark 9.4. A 3-manifold which is an isometric quotient of S 3 or T 3 is a Seifert 3-manifold [52,Section 4]. The analogous statement for orbifolds is false [21].
Proof. We follow the method of proof of [36]. The basic strategy is to construct a partition of O into pieces whose topology can be recognized. Many of the arguments in [36], such as the stratification, are based on the underlying Alexandrov space structure. Such arguments will extend without change to the orbifold setting. Other arguments involve smoothness, which also makes sense in the orbifold setting. We now mention the relevant places in [36] where manifold smoothness needs to be replaced by orbifold smoothness.
• The critical point theory in [36,Section 3.4] can be extended to the orbifold setting using the results in Subsection 2.6. • The results about the topology of nonnegatively curved manifolds in [36,Lemma 3.11] can be extended to the orbifold setting using Lemma 3.20 and Proposition 5.6. • The smoothing results of [36, Section 3.6] can be extended to the orbifold setting using Lemma 2.20 and Corollary 2.21. • Each connected component of O 0−stratum is diffeomorphic either to a closed nonnegatively curved 3-dimensional orbifold, or to the unit disk bundle in the normal bundle of a soul in a complete connected noncompact nonnegatively curved 3-dimensional orbifold. • Each connected component of O slim is the total space of an orbibundle whose base is S 1 or I, and whose fiber is a spherical or Euclidean orientable compact 2-orbifold. • Each connected component of O edge is the total space of an orbibundle whose base is S 1 or I, and whose fiber is D 2 (k) or D 2 (2, 2). In order to prove the proposition, we now follow the method of proof of [36,Section 15].
Each connected component of O 0−stratum has boundary which is empty, a spherical 2orbifold or a Eucldean 2-orbifold. By Proposition 5.6, if the boundary is empty then the component is diffeomorphic to a finite isometric quotient of S 1 × S 2 , S 3 or T 3 . In the S 1 × S 2 case, O is a Seifert orbifold [20, p. 70-71]. Hence we can assume that the boundary is nonempty. By Lemma 3.20, if the boundary is a spherical 2-orbifold then the component is diffeomorphic to D 3 //Γ or I × Z 2 (S 2 //Γ). We group together such components as O 0−stratum Sph . By Lemma 3.20 again, if the boundary is a Euclidean 2-orbifold then the component is diffeomorphic to S 1 × D 2 , S 1 × D 2 (k), S 1 × Z 2 D 2 , S 1 × Z 2 D 2 (k) or I × Z 2 (T 2 //Γ). We group together such components as O 0−stratum Euc . If a connected component of O slim fibers over S 1 then O is closed and has a geometric structure based on R 3 , R × S 2 , Nil or Sol [20, p. 72]. If the structure is R × S 2 or Nil then O is a Seifert orbifold [20, Theorem 1]. If the structure is Sol then O can be cut along a fiber to see that it is a weak graph orbifold. Hence we can assume that each component of O slim fibers over I. We group these components into O slim Sph and O slim Euc , where the distinction is whether the fiber is a spherical 2-orbifold or a Euclidean 2-orbifold.  Furthermore, if Y i is Euclidean then A i = ∅ unless Y i = S 2 (2, 2, 2, 2), in which case A i consists of two D 2 (2, 2)'s. If Y i is spherical then the possibilities are 1. Y i = S 2 and A i consists of two disks D 2 . 2. Y i = S 2 (k, k) and A i consists of two D 2 (k)'s. 3. Y i = S 2 (2, 2, k) and A i consists of D 2 (2, 2) and D 2 (k). ∩ O slim Sph = ∅. By Lemmas 9.5 and 9.6, each of their boundary components is one of S 2 , S 2 (k, k) and S 2 (2, 2, k).

∪ O slim
Euc whose boundary components are S 2 (2, 3, 6), S 2 (2, 4, 4) or S 2 (3, 3, 3). They cannot intersect any other strata, so if there is one such connected component then O is formed entirely of such components. In this case O is diffeomorphic to the result of gluing together two copies of I × Z 2 (T 2 //Γ). Hence O fibers over S 1 //Z 2 and has a geometric structure based on R 3 , Nil or Sol [20, p. 72]. If the structure is Nil then O is a Seifert orbifold [20,Theorem 1]. If the structure is Sol then we can cut O along a generic fiber to see that it is a weak graph orbifold. Hence we can assume that there are no connected components of O 0−stratum Euc whose boundary components are S 2 (2, 3, 6), S 2 (2, 4, 4) or S 2 (3, 3, 3) We cut each such component along { 1 2 } × O ′ and glue on two discal caps. If X 2 is the ensuing orbifold then X 1 is the result of performing a 0-surgery on X 2 , so it suffices to prove that X 2 satisfies the conclusion of the proposition. Therefore we assume henceforth that O slim Sph = ∅.

A remaining connected component of O slim
Euc is diffeomorphic to I × O ′ , where O ′ = S 2 (2, 2, 2, 2). It intersects O edge in four copies of D 2 (2, 2). We cut the connected component of O slim Euc along { 1 2 }×O ′ . The result is two copies of I ×O ′ , each with one free boundary component and another boundary component which intersects O edge in two copies of D 2 (2, 2). If the result X 3 of all such cuttings satisfies the conclusion of the proposition then so does X 2 , it being the result of gluing Euclidean boundary components of X 3 together.
A connected component C of O edge fibers over I or S 1 . Suppose that it fibers over S 1 . Then it is diffeomorphic to S 1 × D 2 (k) or S 1 × D 2 (2, 2), or else is the total space of a bundle over S 1 with holonomy that interchanges the two singular points in a fiber D 2 (2, 2); this is because the mapping class group of D 2 (2, 2) is a copy of Z 2 , as follows from [23,Proposition 2.3]. If C is diffeomorphic to S 1 × D 2 (k) or S 1 × D 2 (2, 2) then it is clearly a weak graph orbifold. In the third case, |C| is a solid torus and the singular locus consists of a circle labeled by 2 that wraps twice around the solid torus. See Figure 10. We can decompose C as C = (S 1 × Z 2 D 2 ) ∪ S 2 (2,2,2,2) C 1 , where C 1 = S 1 × Z 2 (S 2 − 3B 2 ) with one B 2 being sent to itself by the Z 2 -action and the other two B 2 's being switched. See Figure 11. As C 1 is a Seifert orbifold, in any case C is a weak graph orbifold. Put X 4 = X 3 − int(C). If we can show that X 4 is a weak graph orbifold then it follows that X 3 is a weak graph orbifold. Hence we can assume that each connected component of O edge fibers over I.   Each edge corresponds to a copy of If a vertex is of type I ×S 2 (2, 2, 2, 2) then the edge orbifolds only intersect the vertex orbifold on a single one of its two boundary components. Note that |Z| is a solid torus with a certain number of balls removed. Suppose that there are no edges of type I × D 2 (2, 2). Then Z is I × D 2 or I × D 2 (k), which is a weak graph orbifold. Now suppose that there is an edge of type I × D 2 (2, 2). We build up a skeleton for Z. First, the orbifold corresponding to a graph with a single vertex of type I × S 2 (2, 2, 2, 2), and a single edge of type I × D 2 (2, 2), can be identified as the Seifert orbifold C 1 = S 1 × Z 2 (S 2 − 3B 2 ) of before. Let C m be the orbifold corresponding to a graph with m vertices of type I × S 2 (2, 2, 2, 2) and m edges of type I × D 2 (2, 2). See Figure 12. Then C m is an m-fold cover of C 1 and is also a Seifert orbifold.
Returning to the orbifold Z, there is some m so that Z is diffeomorphic to the result of starting with C m and gluing some S 1 × Z 2 D 2 (k i )'s onto some of the boundary S 2 (2, 2, 2, 2)'s, where k i ≥ 1. See Figure 13 for an illustrated example.
Thus Z is a weak graph orbifold.
As X 3 is the result of gluing Z to a circle bundle over a surface, X 3 is a weak graph orbifold. Along with Proposition A.10, this proves the proposition.
Proposition 9.7. Let c 3 be the volume of the unit ball in R 3 , let K ≥ 10 be a fixed integer and let N be a positive integer. Fix a function A : (0, ∞) → (0, ∞). Then there is a w 0 ∈ (0, c 3 /N) such that the following holds.
Suppose that (O, g) is a compact connected orientable Riemannian 3-orbifold with boundary. Assume in addition that (2) The diameters of the connected components of ∂O are bounded above by w 0 .
Then O is diffeomorphic to • The result of performing 0-surgeries on a strong graph orbifold, Proof. We follow the method of proof of [35,Section 16]. The effective difference from the proof of Proposition 9.1 is that we have additional components of O 0−stratum , which are diffeomorphic to I × (T 2 //Γ). If such a component is diffeomorphic to I × T 2 or I ×S 2 (2, 2, 2, 2) then we can incorporate it into the weak graph orbifold structure. The other cases give rise to the additional possibilities listed in the conclusion of the proposition. 10. Incompressibility of cuspidal cross-sections and proof of Theorem 1.1 In this section we complete the proof of Theorem 1.1.
With reference to Proposition 8.9, given a sequence t α → ∞, let Y α be the truncation of N i=1 H i obtained by removing horoballs at distance approximately 1 2β(t α ) from the basepoints Proof. The proof is similar to that of [36,Theorem 17.3].
So far we know that if α is large then the 3-orbifold O t α has a (possibly empty) hyperbolic piece whose complement satisfies the conclusion of Proposition 9.7. In this section we show that there is such a decomposition of O t α so that the hyperbolic cusps, if any, are incompressible in O t α .
The corresponding manifold result was proved by Hamilton in [32] using minimal disks. He used results of Meeks-Yau [40] to find embedded minimal disks with boundary on an appropriate cross-section of the cusp. The Meeks-Yau proof in turn used a tower construction [41] similar to that used in the proof of Dehn's Lemma in 3-manifold topology. It is not clear to us whether this line of proof extends to three-dimensional orbifolds, or whether there are other methods using minimal disks which do extend. To circumvent these issues, we use an alternative incompressibility argument due to Perelman [48,Section 8.2] that exploits certain quantities which change monotonically under the Ricci flow. Perelman's monotonic quantity involved the smallest eigenvalue of a certain Schrödinger-type operator. We will instead use a variation of Perelman's argument involving the minimal scalar curvature, following [35,Section 93.4].
Before proceeding, we need two lemmas: Proof. If a 0-surgery adds a neck (S 2 //Γ) × I then we can put a metric on the neck which is an isometric quotient of a slight perturbation of the doubled Schwarzschild metric [2, (1.23)] on S 2 × I. Hence we can perform the 0-surgery so that the scalar curvature is bounded below by − 3 2 + ǫ 10 and the volume increases by at most ǫ 10 ; see [2, p. 155] and [49] for the analogous result in the manifold case. The lemma now follows from an overall rescaling to make R ≥ − 3 2 . Lemma 10.3. Suppose that O is a strong graph orbifold with boundary components C 1 , . . . , C k . Let H 1 , . . . , H k be truncated hyperbolic cusps, where ∂H i is diffeomorphic to C i for all i ∈ {1, . . . , k}. Then for all ǫ > 0, there is a metric on O with scalar curvature ≥ − 3 2 such that vol(O) < ǫ, and C i has a collar which is isometric to one side of a collar neighborhood of a cuspical 2-orbifold in H i .
Proof. We first prove the case when O is a closed strong graph manifold. The strong graph manifold structure gives a graph whose vertices {v a } correspond to the Seifert blocks and whose edges {e b } correspond to 2-tori. For each vertex v a , let M a be the corresponding Seifert block. We give it a Riemannian metric g a which is invariant under the local S 1actions and with the property that the quotient metric on the orbifold base is a product near its boundary. Then g a has a product structure near ∂M a . Given δ > 0, we uniformly shrink the Riemannian metric on g a by δ in the fiber directions. As δ → 0, the volume of M a goes to zero while the curvature stays bounded.
Let T 2 b be the torus corresponding to the edge e b . There are associated toral boundary components {B 1 , B 2 } of Seifert blocks. Given δ > 0 and i ∈ {1, 2}, consider the warped product metric ds 2 + e −2s g B i on a product manifold P δ,i = [0, L δ,i ] × B i . We attach this at B i to obtain a C 0 -metric, which we will smooth later. The sectional curvatures of P δ,i are −1 and the volume of P δ,i is bounded above by the area of B i . We choose L δ,i so that the areas of the cross-sections {L δ,1 } × B 1 and {L δ,2 } × B 2 are both equal to some number A. Finally, consider R 3 with the Sol-invariant metric e −2z dx 2 + e 2z dy 2 + dz 2 . Let Γ be a Z 2 -subgroup of the normal R 2 -subgroup of Sol. Note that the curvature of R 3 /Γ is independent of Γ. The z-coordinate gives a fibering z : R 3 /Γ → R with T 2 -fibers. We can choose Γ = Γ δ and an interval [c 1 , can be taken independent of A. We attach z −1 ([c 1 , c 2 ]) to the previously described truncated cusps, at the boundary components {L δ,1 } × B 1 and {L δ,2 } × B 2 . See Figure 14. Taking A sufficiently small we can ensure that (10.4) vol(P δ,1 ) + vol(P δ,2 ) + vol(z −1 ([c 1 , c 2 ])) < area(B 1 ) + area(B 2 ) + δ.
We repeat this process for all of the tori {T 2 b }, to obtain a piecewise-smooth C 0 -metric g δ on O.
As δ → 0, the sectional curvature stays uniformly bounded on the smooth pieces. Furthermore, the volume of (O, g δ ) goes to zero. By slightly smoothing g δ and performing an overall rescaling to ensure that the scalar curvature is bounded below by − 3 2 , if δ is sufficiently small then we can ensure that vol(O, g δ ) < ǫ. This proves the lemma when O is a closed strong graph manifold.
If O is a strong graph manifold but has nonempty boundary components, as in the hypotheses of the lemma, then we treat each boundary component C i analogously to a factor B 1 in the preceding construction. That is, given parameters 0 < c 1,C i < c 2,C i , we start by putting a truncated hyperbolic metric ds 2 + e −2s g ∂H i on [c 1,C i , c 2,C i ] × C i . This will be the metric on the collar neighborhood of C i , where {c 1,C i } × C i will end up becoming a boundary component of O. We take c 2,C i so that the area of {c 2,C i } × C i matches the area of a relevant cross-section of the truncated cusp extending from a boundary component B 2,i of a Seifert block. We then construct a metric g δ on O as before. If we additionally take the parameters {c 1,C i } sufficiently large then we can ensure that vol(O, g δ ) < ǫ.
Finally, if O is a strong graph orbifold then we can go through the same steps. The only additional point is to show that elements of the (orientation-preserving) mapping class group of an oriented Euclidean 2-orbifold T 2 //Γ are represented by affine diffeomorphisms, in order to apply the preceding construction using the Sol geometry. To see this fact, if Γ is trivial then the mapping class group of T 2 is isomorphic to SL(2, Z) and the claim is clear. To handle the case when T 2 //Γ is a sphere with three singular points, we use the fact that the mapping class group of a sphere with three marked points is isomorphic to the permutation group of the three points [23,Proposition 2.3]. The mapping class group of the orbifold T 2 //Γ will then be the subgroup of the permutation group that preserves the labels. If T 2 //Γ is S 2 (2, 3, 6) then its mapping class group is trivial. If T 2 //Γ is S 2 (2, 4, 4) then its mapping class group is isomorphic to Z 2 . Picturing S 2 (2, 4, 4) as two right triangles glued together, the nontrivial mapping class group element is represented by the affine diffeomorphism which is a flip around the "2" vertex that interchanges the two triangles. If T 2 //Γ is S 2 (3, 3, 3) then its mapping class group is isomorphic to S 3 . Picturing S 2 (3, 3, 3) as two equilateral triangles glued together, the nontrivial mapping class group elements are represented by affine diffeomorphisms as rotations and flips. Finally, if T 2 //Γ is S 2 (2, 2, 2, 2) then its mapping class group is isomorphic to PSL (  By assumption, the components of Y satisfy the conclusion of Proposition 9.7. Hence Y has a decomposition into connected components Y = Y 0 ⊔. . .⊔Y n , where Y 0 is the component containing C 0 , and Y 0 arises from a strong graph orbifold by 0-surgeries, as otherwise there would not be a compressing discal orbifold. By Lemma A.11, Y 0 comes from a disjoint union A ⊔ B via 0-surgeries, where A is one of the four possibilities of that Lemma, and B is a strong graph orbifold. By Lemmas 10.2 and 10.3, we may assume without loss of generality that B = ∅. To construct the desired metric on O ′ , we proceed as follows. Let H 0 , . . . , H n be the cusps of the hyperbolic orbifold N, where H 0 corresponds to the component C 0 of Y . We first truncate N along totally umbilic cuspical 2-orbifolds C 0 , . . . , C n . Pick ǫ > 0. For each i ≥ 1 such that the component Y i comes from 0-surgeries on a strong graph orbifold, we use Lemmas 10.2 and 10.3 to find a metric with R ≥ − 3 2 on Y i , which glues isometrically along the corresponding cusps in C 1 ⊔ . . . ⊔ C n , and which can be arranged to have volume < ǫ by taking the C i 's to be deep in their respective cusps. For the components Y i , i ≥ 1, which do not come from a strong graph orbifold via 0-surgery, we may also find metrics with R ≥ − 3 2 and arbitrarily small volume, which glue isometrically onto the corresponding truncated cusps of N (when they have nonempty boundary). Our final step will be to find a metric on Y 0 = A with R ≥ − 3 2 which glues isometrically to C 0 , and has volume strictly smaller than the portion of the cusp H 0 cut off by C 0 . Since ǫ is arbitrary, this will yield a contradiction.
If A is S 1 × Z 2 D 2 or S 1 × Z 2 D 2 (k) we can perform the construction of the previous paragraph equivariantly with respect to the Z 2 -action, to form the desired metric on S 1 × Z 2 D 2 (or S 1 × Z 2 D 2 (k)).
Proof of Theorem 1.1 : Lemma 7.28 implies that for large α, the original 3-orbifold O is the result of performing 0-surgeries on a disjoint union of O t α and a finite number of isometric quotients of S 1 × S 2 and S 3 . From Proposition 10.8, O t α can be split along incompressible Euclidean 2-orbifolds so that for each remaining piece P, either • P admits a finite-volume complete hyperbolic metric, or • Its metric closure P satisfies the conclusion of Proposition 9.7.
Referring to Proposition 9.7, if P is an isometric quotient of S 3 or T 3 then it already has a geometric structure. If P is I × S 2 (p, q, r) with 1 p + 1 q + 1 r = 1 then we can remove it without losing any information. If P is I × Z 2 S 2 (p, q, r) with 1 p + 1 q + 1 r = 1 then P has a Euclidean structure.
Finally, suppose that P is the result of performing 0-surgeries on a collection of strong graph orbifolds in the sense of Definition A.7. A Seifert-fibered 3-orbifold with no bad 2dimensional suborbifolds is geometric in the sense of Thurston [5,Proposition 2.13]. This completes the proof of Theorem 1.1.
Remark 10.9. The geometric decomposition of O that we have produced, using strong graph orbifolds, will not be minimal if O has Sol geometry. In such a case, O fibers over a 1dimensional orbifold. Cutting along a fiber and taking the metric completion gives a product orbifold, which is a graph orbifold. Of course, the minimal geometric decomposition of O would leave it with its Sol structure.
Remark 10.10. Theorem 1.1 implies that O is very good, i.e. the quotient of a manifold by a finite group action [4,Corollary 1.3]. Hence one could obtain the geometric decomposition of O by running Perelman's proof equivariantly, as is done in detail for elliptic and hyperbolic manifolds in [19]. However, one cannot prove the geometrization of orbifolds this way, as the reasoning would be circular; one only knows that O is very good after proving Theorem 1.1.

Appendix A. Weak and strong graph orbifolds
In this appendix we provide proofs of some needed facts about graph orbifolds. We show that a weak graph orbifold is the result of performing 0-surgeries on a strong graph orbifold. (Since we don't require strong graph orbifolds to be connected, we need only one.) In order to clarify the arguments, we prove the corresponding manifold results before proving the orbifold results. Here "metric completion" refers to an arbitrary Riemannian metric on M.
Remark A.2. In the definition of a weak graph manifold, we could have instead required that the connected components of the metric completion of M − i T i are circle bundles over surfaces. This would give an equivalent notion, since any Seifert-fibered 3-manifold can be cut along tori into circle bundles over surfaces. Proof. Suppose that a connected strong graph manifold M contains an essential embedded 2-sphere S. We can assume that S is transverse to {T i }. We choose S among all such essential embedded 2-spheres so that the number of connected components of S ∩ i T i is as small as possible.
If S ∩ i T i = ∅ then S is an essential 2-sphere in one of the Seifert components.
If S ∩ i T i = ∅, let C be an innermost circle in S ∩ i T i . Then C ⊂ T k for some k and C = ∂D for some 2-disk D embedded in a Seifert component U with T k ⊂ ∂U. As T k is incompressible, C = ∂D ′ for some 2-disk D ′ ⊂ T k . If D ∪ D ′ bounds a 3-ball in U then we can isotop S to remove the intersection with T k , which contradicts the choice of S. Thus D ∪ D ′ is an essential 2-sphere in U.
In any case, we found an essential 2-sphere in one of the Seifert pieces. It follows that the Seifert piece, and hence all of M, is diffeomorphic to S 1 × S 2 or RP 3 #RP 3 [52, p. 432].
Proposition A.5. A weak graph manifold is a connected sum of strong graph manifolds.
Proof. We start with a compact orientable 3-manifold M, along with a collection of disjoint embedded 2-tori, satisfying condition (1) of Definition A.3 but maybe not conditions (2) and (3). We introduce two operations.
First operation : We look for a torus T j for which the two induced circle fibrations (coming from the adjacent Seifert bundles) are isotopic. If there is one, we extend the Seifert fibration over T j and remove T j from the collection. This terminates the first operation.
Second operation : We look for a torus T j which is compressible. If there is one, let M ′ be the connected component of M containing T j . Let D be a compressing disk, which we can assume to be transversal to {T i }. We choose such a compressing disk so that D ∩ i T i has the smallest possible number of connected components. Let C be an innermost circle in D ∩ i T i . Then C bounds a disk D ′ lying in one of the Seifert bundles, say with boundary T k .
If C also bounds a disk D ′′ ⊂ T k then D ′ ∪ D ′′ is an embedded 2-sphere S in the Seifert bundle. If S is not essential in the Seifert bundle then we can isotop D so that it does not intersect T k , which contradicts the choice of D. So S is essential in the Seifert bundle. Then the Seifert bundle, and hence all of M ′ , is diffeomorphic to S 1 × S 2 or RP 3 #RP 3 . In this case we remove M ′ , which is a strong graph manifold, and thereby reduce the number of tori.
Thus we can assume that C is a compressing disk for one of the Seifert bundles, which is necessarily a solid torus S 1 × D 2 [52, Corollary 3.3].
Let U be the Seifert bundle on the other side of T k from S 1 × D 2 . Let B be the orbifold base of U, with projection π : U → B. There is a circle boundary component R ⊂ ∂B so that T k = π −1 (R). That is, the solid torus S 1 × D 2 is glued to U along π −1 (R).
If ∂D 2 ⊂ D 2 ⊂ S 1 × D 2 is not isotopic to a fiber of π T k , let u > 0 be their algebraic intersection number in T k . Then U ∪ T k (S 1 ×D 2 ) has a Seifert fibration over B ∪ R D 2 (u). We remove T k , thereby reducing the number of tori, and go to the End of the Second Operation.
Suppose that ∂D 2 ⊂ D 2 ⊂ S 1 × D 2 is isotopic to a fiber of π T k . We identify S 1 × D 2 with R × D 2 .
Step 1 : We look for a smooth embedded curve γ : [0, 1] → B whose interior lies in the regular part of B, with γ(0), γ(1) ∈ R, so that γ separates B into compact orbifolds B 1 and B 2 neither of which is a nonsingular 2-disk. See Figure 15.
If there is such a curve, put i ) inherits a weak graph structure from M ′ . This shows that M i is a weak graph manifold.
If Y 1 = D 3 and Y 2 = D 3 then we have found a nontrivial connect sum decomposition M ′ = M 1 #M 2 . We replace M ′ by M 1 ⊔ M 2 , a weak graph manifold (with one more torus than M ′ ). If Y 1 = Y 2 = D 3 then M ′ = S 3 , which is a strong graph manifold, and we remove it. If exactly one of Y 1 and Y 2 is diffeomorphic to D 3 , suppose that it is Y 2 . Note that in this case ∂B 2 must contain a boundary component of B. Now M ′ is diffeomorphic to M 1 . In this way we found a weak graph decomposition of M ′ with fewer tori. We now repeat Step 1 as many times as possible.
Step 2 : Next, we suppose that there is no curve γ as in Step 1, but there is a nonseparating smooth curve γ : [0, 1] → B whose interior lies in the regular part of B, with γ(0), γ(1) ∈ R. See Figure 16.
where D 3 1 and D 3 2 are 3-disks and D 3 i intersects R i × D 2 . Then M ′ = M 1 #(S 1 × S 2 ). We claim that M 1 is a weak graph manifold. To see this, note that (R i × D 2 ) ∪ D 3 i is a solid torus. Also, M 1 − ((R × D 2 ) ∪ D 3 1 ∪ D 3 2 ) inherits a weak graph structure from M ′ . Hence, M 1 is a weak graph manifold (with one more torus than M ′ ). We replace M ′ by M 1 . We repeat Step 2 as many times as possible.
Step 3 : Finally, if there is no curve γ as in Steps 1 and 2 then B is S 1 × I, D 2 or D 2 (r). If B is S 1 × I then π −1 (B) ∪ T k (S 1 × D 2 ) is diffeomorphic to S 1 × D 2 , so we can remove T k and reduce the number of tori. If B is D 2 or D 2 (r) then M ′ is S 3 or a lens space, which is a strong graph manifold. In these cases we remove M ′ , thereby reducing the number of tori.
End of the Second Operation : The outcome so far of the second operation is that we either found a nontrivial connected sum decomposition of M ′ = # s M s in terms of weak graph manifolds M s , along possibly with some strong graph manifolds that we threw away, or we found a weak graph decomposition of M with fewer tori. In the first case we replace M ′ by s M s . In the second case we use the new weak graph decomposition of M. This terminates the second operation.
Starting with the weak graph manifold M, we perform the first operation as many times as possible, then the second operation as many times as possible, then the first operation as many times as possible, then the second operation as many times as possible, etc. This process must stop, because of the finiteness of the prime decomposition of the connected components of M and the ensuing bound on the number of tori.
We have produced a decomposition of each component of M as either a strong graph manifold of a special type or as a connected sum of certain weak graph manifolds. As we can no longer perform the operations, each of these weak graph manifolds must be a strong graph manifold. This proves the theorem.
A.2. Weak graph orbifolds are connected sums of strong graph orbifolds. In this section we only consider 3-dimensional orbifolds that do not admit embedded bad 2-dimensional suborbifolds. From Subsection 2.4, each E i is diffeomorphic to T 2 or S 2 (2, 2, 2, 2).
Proposition A.8. If a connected strong graph orbifold contains an essential embedded spherical 2-orbifold then it is diffeomorphic to a finite isometric quotient of S 1 × S 2 .
Proof. Suppose that a connected strong graph orbifold O contains an essential embedded spherical 2-orbifold S. Lemma A.9. After an isotopy of S, we can assume that S ∩ i E i is a disjoint collection of closed curves in the regular part of S.
Proof. If E i is diffeomorphic to T 2 then a neighborhood of E i lies in |O| reg and after isotopy, S ∩ E i is a disjoint collection of closed curves in the regular part of S. Suppose that E i = S 2 (2, 2, 2, 2). A neighborhood of E i is diffeomorphic to I × E i . Suppose that p ∈ S intersects a singular point of E i . Then the local group of p in S must be Z 2 . After pushing a neighborhood of p ∈ S slightly in the I-direction of I × E i , we can remove the intersection of S with that particular singular point of E i . In this way, we can arrange so that S intersects i E i transversely, with the intersection lying in the regular part of S.
We choose S among all such essential embedded spherical 2-orbifolds so that the number of connected components of |S ∩ i E i | is as small as possible.
If S ∩ i E i = ∅ then S is an essential embedded spherical 2-orbifold in one of the Seifert pieces.
Then D ∪ D ′ is an embedded 2-orbifold with underlying space S 2 and at most two singular points. As O has no bad 2-suborbifolds, D ∪ D ′ must be S 2 (r, r) for some r ≥ 1. If D ∪ D ′ bounds some D 3 (r, r) in U then we can isotop S to remove the intersection with E k , which contradicts the choice of S. Thus D ∪ D ′ is an essential embedded spherical 2-orbifold in U.
In any case, we found an essential embedded spherical 2-orbifold in one of the Seifert pieces. Then the universal cover of the Seifert piece contains an essential embedded S 2 . It follows that the universal cover of the Seifert piece is R × S 2 [5, Proposition 2.13]. The Seifert piece, and hence all of O, must then be diffeomorphic to a finite isometric quotient of S 1 × S 2 .
Proposition A.10. A weak graph orbifold is the result of performing 0-surgeries on a strong graph orbifolds.
Proof. We start with a compact orientable 3-orbifold O, along with a collection {E i } of disjoint embedded orientable Euclidean 2-orbifolds, satisfying condition (1) of Definition A.7 but maybe not conditions (2) and (3). We introduce two operations.
First operation : We look for a 2-orbifold E j for which the two induced circle fibrations (coming from the adjacent Seifert bundles) are isotopic, in the sense of [5,Chapter 2.5]. If there is one, we extend the Seifert fibration over E j and remove E j from the collection. This terminates the first operation.
Second operation : We look for a 2-orbifold E j which is compressible. If there is one, let O ′ be the connected component of O containing E j . Let D be a compressing discal 2-orbifold. As in Lemma A.9, we can assume that D intersects i E i transversally, with the intersection lying in the regular part of D. We choose such a compressing discal 2-orbifold so that D ∩ i E i has the smallest possible number of connected components. Let C be an innermost circle in D ∩ i E i . Then C bounds a discal 2-orbifold D ′ lying in one of the Seifert bundles, say with boundary E k .
If C also bounds a discal 2-orbifold D ′′ ⊂ E k then D ′ ∪ D ′′ is an embedded 2-orbifold in the Seifert bundle. As there are no bad 2-orbifolds in O, we must have D ′ ∪ D ′′ = S 2 (r, r) for some r ≥ 1. If this S 2 (r, r) is not essential in the Seifert bundle then it bounds a D 3 (r, r) in the Seifert bundle. In this case we can isotop D so that it does not intersect E k , which contradicts the choice of D. So the S 2 (r, r) is essential in the Seifert bundle. Then as in Proposition A.8, the Seifert bundle (and hence all of O ′ ) is diffeomorphic to a finite isometric quotient of S 1 ×S 2 . In this case we remove O ′ , which is a strong graph orbifold, and thereby reduce the number of tori.
Thus we can assume that C is a compressing discal 2-orbifold for one of the Seifert bundles, call it U ′ , which is necessarily a solid-toric orbifold S 1 × D 2 (r) or S 1 × Z 2 D 2 (r) for some r ≥ 1 [17,Lemma 2.47].
Let U be the Seifert bundle on the other side of E k from U ′ . Let B be the orbifold base of U, with projection π : U → B. There is a 1-orbifold boundary component R ⊂ ∂B, diffeomorphic to S 1 or S 1 //Z 2 , so that E k = π −1 (R). That is, U ′ is glued to S along π −1 (R). There is also a fibering σ : U ′ → R (which is not an Seifert fibering).
We refer to [5,Chapter 2.4] for a discussion of Dehn fillings, i.e. gluings of U ′ to π −1 (R). If the meridian curve of U ′ is not isotopic to a fiber of π E k , let u > 0 be the algebraic intersection number (in the maximal abelian subgroup of π 1 (E k )). Then the gluing of U ′ to U, along π −1 (R), has a Seifert fibration. We remove E k , thereby reducing the number of tori and go to the End of the Second Operation.
Suppose now that the meridian curve is isotopic to a fiber of π.
If Y 1 = D 3 (r, r) and Y 2 = D 3 (r, r) then we have found a nontrivial connect sum decomposition O ′ = O 1 # S 2 (r,r) O 2 . We replace O ′ by O 1 ⊔ O 2 . If Y 1 = Y 2 = D 3 (r, r) then M ′ = S 3 //Z r is a strong graph orbifold and we remove it. If exactly one of Y 1 and Y 2 is diffeomorphic to D 3 (r, r), suppose that it is Y 2 . Note that in this case, ∂B 2 must contain a boundary component of B. Now O ′ is diffeomorphic to O 1 . In this way we found a weak graph decomposition of O ′ with fewer Euclidean 2-orbifolds. We now repeat Step 1 as many times as possible.
If Y 1 = D 3 (2, 2, r) and Y 2 = D 3 (2, 2, r) then we found a nontrivial connected sum decomposition O ′ = O 1 # S 2 (2,2,r) O 2 . We replace O ′ by O 1 ⊔ O 2 , a weak graph orbifold (with one more Euclidean 2-orbifold than O ′ ). If Y 1 = Y 2 = D 3 (2, 2, r) then O ′ = S 3 //D r , which is a strong graph orbifold, and we remove it. If exactly one of Y 1 and Y 2 is D 3 (2, 2, r), suppose that it is Y 2 . Note that in this case ∂B 2 must contain a boundary component of B. Now O ′ is diffeomorphic to O 1 . In this way we found a weak graph decomposition of O ′ with fewer Euclidean 2-orbifolds. We now repeat Step 3 as many times as possible.
Step 5 : Finally, if there is no such curve γ as in Steps 1, 2, 3 and 4 then we have the following possibilities for B.
(1) B = S 1 × I. In this case π −1 (B) ∪ E k (S 1 × D 2 (r)) is diffeomorphic to S 1 × D 2 (r), so we can remove E k and reduce the number of Euclidean 2-orbifolds. (2) B = D 2 . Then π −1 (B) ∪ E k (S 1 × D 2 (r)) is diffeomorphic to S 3 //Z r . Its underlying space is S 3 and the singular locus is a circle labelled by r. In this case O ′ is a strong graph orbifold and we remove it, thereby removing E k . (3) B = D 2 (s). Then π −1 (B) ∪ E k (S 1 × D 2 (r)) is diffeomorphic to S 3 //(Z r × Z s ).
Its underlying space is S 3 and the singular locus consists of two linked circles, one labelled by r and the other labelled by s. In this case O ′ is a strong graph orbifold and we remove it, thereby removing E k . (4) B = D 2 //Z 2 . Then π −1 (B) ∪ E k (S 1 × Z 2 D 2 (r)) has a double cover (S 1 × D 2 ) ∪ T 2 (S 1 × D 2 (r)) which is diffeomorphic to S 3 //Z r , from case (2) above. Taking the Z 2 -quotient shows that π −1 (B) ∪ E k (S 1 × Z 2 D 2 (r)) is diffeomorphic to S 3 //D r . The singular locus in S 3 consists of a circle labelled by 2 and an arc joining two points in the circle, labelled by r. In this case O ′ is a strong graph orbifold and we remove it, thereby removing E k . (5) B = D 2 //D s . Then π −1 (B) ∪ E k (S 1 × Z 2 D 2 (r)) has a double cover (S 1 × D 2 (s)) ∪ T 2 (S 1 × D 2 (r)) which is diffeomorphic to S 3 //(Z r × Z s ) from case (3) above. Taking the Z 2 -quotient shows that π −1 (B) ∪ E k (S 1 × Z 2 D 2 (r)) is diffeomorphic to (S 3 //(Z r × Z s ))//Z 2 . The singular locus in S 3 consists of a circle labelled by 2 and two arcs joining points in the circle, labelled by r and s, with the four endpoints of the arcs interleaved on the circle. In this case O ′ is a strong graph orbifold and we remove it, thereby removing E k .  The underlying space of π −1 (B) ∪ E k (S 1 × Z 2 D 2 (r)) is a solid torus. The singular locus consists of a contractible circle labelled by 2 and an arc joining two points in the circle, labelled by r (if r > 1), so that if the "2" circle were shrunk to a point then the result would be a core curve in the solid torus. See Figure 20.
End of the Second Operation : The outcome so far is that we either found a nontrivial representation of O ′ as the result of performing 0-surgeries on some weak graph orbifolds O s and possibly some additional strong graph orbifolds, or we found a weak graph decomposition of O with fewer Euclidean 2-orbifolds. In the first case we replace O ′ by s O s . In the second case we use the new weak graph decomposition of O. This terminates the second operation.
Starting with the weak graph orbifold O, we perform the first operation as many times as possible, then the second operation as many times as possible, then the first operation We have produced a decomposition of each component of O as either a strong graph orbifold of a special type or as the result of performing 0-surgeries on certain weak graph orbifolds. As we can no longer perform the operations, each of these weak graph orbifolds must be a strong graph orbifold. This proves the theorem. • O i is a strong graph manifold for all i.
Proof. Let Z be a compressing discal orbifold for C.
By Proposition A.10 we know that O comes from 0-surgery on a collection O 0 , . . . , O n of strong graph orbifolds, where ∂O 0 contains C. Consider a collection C = {S 1 , . . . , S k } ⊂ O of spherical 2-suborbifolds associated with such a 0-surgery description of O. We may assume that Z is transverse to C, and that the number of connected components in the intersection Z ∩ C is minimal among such compressing discal orbifolds. Reasoning as in the proof of Lemma A.9, we conclude that Z is disjoint from C. Therefore after cutting O open along C and filling in the boundary components to undo the 0-surgeries, we get that Z lies in O 0 . Similar reasoning shows that Z must lie in a single Seifert component U of O 0 . An orientable Seifert 3-orbifold with a compressible boundary component must be a solid-toric 3-orbifold [17, Lemma 2.47]. The lemma follows.