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Compact foliations with finite transverse LS category

We prove that if F is a foliation of a compact manifold M with all leaves compact submanifolds, and the transverse saturated category of F is finite, then the leaf space M/F is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. Colman proved in her 1998 doctoral thesis that the transverse saturated category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with finite transverse saturated category.


Introduction
A compact foliation is a foliation of a manifold M with all leaves compact submanifolds. For codimension one or two, a compact foliation F of a compact manifold M defines a fibration of M over its leaf space M/F which is an orbifold [27,11,12,33,10]. For codimension three and above, the leaf space M/F of a compact foliation need not be a Hausdorff space. This was first shown by an example of Sullivan [30] of a flow on a 5-manifold whose orbits are circles, and the lengths of the orbits are not bounded above. Subsequent examples of Epstein and Vogt [13,35] showed that for any codimension greater than two, there are examples of compact foliations of compact manifolds whose leaf spaces are not Hausdorff. Vogt gave a remarkable example of a 1-dimensional, compact C 0 -foliation of R 3 with no upper bound on the lengths of the circle leaves in [36]. The results described below apply to the case of compact C 1 -foliations of compact manifolds.
A compact foliation F with Hausdorff leaf space is said to be compact Hausdorff. The holonomy of each leaf of a compact Hausdorff foliation is a finite group, a property which characterizes them among the compact foliations. If every leaf has trivial holonomy group, then a compact Hausdorff foliation is a fibration. Otherwise, a compact Hausdorff foliation is a "generalized Seifert fibration", where the leaf space M/F is a "V-manifold" [29,17,22].
A compact foliation whose leaf space is non-Hausdorff has a closed, non-empty saturated subset, the bad set X 1 ⊂ M , which is the union of the leaves whose holonomy group is infinite. The image of X 1 in the leaf space M/F consists of the non-Hausdorff points for the T 1 topology on M/F. The work by Edwards, Millett and Sullivan [10] established many fundamental properties of the geometry of the leaves of a compact foliation near its bad set, yet there is no general structure theory for compact foliations, comparable to what is understood for compact Hausdorff foliations. The results of § §4, 5 and 6 of this work provide new insights and techniques for the study of these foliations. In particular, we introduce the notion of a tame point for the bad set X 1 in Definition 5.1. This notion is a new property of non-Hausdorff compact foliations, and is studied in Section 5.
The transverse Lusternik-Schnirelmann (LS) category of foliations was introduced in the 1998 thesis of H. Colman [4,8]. The  The transverse category cat ∩ | (M, F) of a compact Hausdorff foliation F of a compact manifold M is always finite [8], as every leaf admits a saturated product neighborhood which is transversely categorical. For a non-Hausdorff compact foliation, our main result is that there is no transversely categorical covering of the the bad set. THEOREM 1.2. Let F be a compact C 1 -foliation of a compact manifold M with non-empty bad set X 1 . Then there exists a dense set of tame points X t 1 ⊂ X 1 . Moreover, for each x ∈ X t 1 , there is no transversely categorical saturated open set containing x. Proof. Millett [22] and Epstein [12] showed that for a compact Hausdorff foliation F of a manifold M each leaf has finite holonomy, and thus M admits a Riemannian metric so that the foliation is Riemannian.
Recall that a foliation is geometrically taut if the manifold M admits a Riemannian metric so that each leaf is an immersed minimal manifold [28,31,15]. Rummler proved in [28] that a compact foliation is Hausdorff if and only if it is taut, and thus we can conclude: COROLLARY 1.4. A compact C 1 -foliation of a compact manifold M with cat ∩ | (M, F) < ∞ is geometrically taut.
The idea of the proof of Theorem 1.2 is as follows. The formal definition of the bad set X 1 in §3 is that it consist of leaves of F such that every open neighborhood of the leaf contains leaves of arbitrarily large volume. This characterization of the bad set intuitively suggests that it should be a rigid set. That is, any foliated homotopy of an open neighborhood of a point in the bad set should preserve these dynamical properties, hence the open neighborhood cannot be continuously retracted to a single leaf. The proof of this statement is surprisingly delicate, and requires a very precise understanding of the properties of leaves in an open neighborhood of the bad set. A key result is Proposition 5.2, an extension of the Moving Leaf Lemma in [10], which establishes the existence of "tame points".
The overview of the paper is as follows: The first two sections consist of background material, which we recall to establish notations, and also present a variety of technical results required in the later sections. In §2 we give some basic results from foliation theory, and in §3 we recall some basic results about compact foliations, especially the structure theory for the good and the bad sets. In §4 we establish a key homological property for compact leaves under deformation by a homotopy. The techniques introduced in this section are used again in later sections. The most technical results of the paper are contained in §5, where we prove that tame points are dense in the bad set. Finally, in §6 we prove that an open saturated set containing a tame point is not categorical. Theorem 1.2 follows immediately from Propositions 5.2 and 6.3.

Foliation preliminaries
We assume that M is a compact smooth Riemannian manifold without boundary of dimension m = p + q, that F is a compact C 1 -foliation of codimension-q, and that the leaves of F are smoothly immersed compact submanifolds, so that F is more precisely a C 1,∞ -foliation. For x ∈ M , denote by L x the leaf of F containing x.
We recall below some well-known facts about foliations, and introduce some conventions of notation. The books [3,14,16] provide excellent basic references; our notation is closest to that used in [3]. Note that the analysis of the bad sets in later sections requires careful estimates on the foliation geometry; not just in each leaf, but also for nearby leaves of a given leaf. This requires a careful description of the local metric geometry of a foliation, as given in this section.
2.1. Tangential and normal geometry. Let T F denote the tangent bundle to F, and let Π : Q → M denote its normal bundle, identified with the subbundle T F ⊥ ⊂ T M of vectors orthogonal to T F. The Riemannian metric on T M induces Riemannian metrics on both T F and Q by fiberwise restriction. For a vector v ∈ T x M , let v denote its length in the Riemannian metric. Then for v ∈ T x F the length in the induced leafwise metric is also denoted by v .
For > 0, let T M ⊂ T M denote the disk subbundle of vectors with length less than , and let T F ⊂ T F and Q ⊂ Q be the corresponding -disk subbundles of T F and Q, respectively.
For a leaf L ⊂ M , let d L : L×L → [0, ∞) be the distance function on L for the restricted Riemannian metric on L. That is, for x, x ∈ L the distance d L (x, x ) is the infimum of the lengths of piece-smooth leafwise paths between x and x . As L is compact, the manifold L with the metric d L is a complete metric space, and the distance d L (x, x ) is realized by a leafwise geodesic path from x to x . We introduce the notation d F for the collection of leafwise distance functions, where d F (x, y) = d L (x, y) if x, y ∈ L, and otherwise d F (x, y) = ∞. Given r > 0 and a set K ⊂ L, let Let exp = exp M : T M → M denote the exponential map for d M which is well-defined as M is compact. For x ∈ M , we let exp M x : T x M → M denote the exponential map at x. For x ∈ L, we let exp F x : T x L → L denote the exponential map for the leafwise Riemannian metric. Then exp F x maps the ball B TxL (0, r) of radius r in T x L onto the set B F (x, r). We next chose 0 > 0 so that it satisfies a sequence of conditions, as follows. For each x ∈ M , the differential It follows that there exists x > 0 such that the restriction exp M x : T x x M → M is a diffeomorphism. As M is compact, there exists 0 > 0 such that for all x ∈ M , the restriction exp M x : T 0 x M → M is a diffeomorphism onto its image. Thus, 0 is less than the injectivity radius of the Riemannian metric on M . (See [1,9] for details of the properties of the injectivity radius of the geodesic map.) We also require that 0 > 0 be chosen so that for all x ∈ M : Let exp Q x : Q x → M denote the restriction of exp M x to the normal bundle at x. Then for all x ∈ M , exp Q x : Q 0 x → M is a diffeomorphism onto its image. We also require that 0 > 0 satisfy: x → M is transverse to F, and that the image exp Q x (Q 0 x ) of the normal disk has angle at least π/4 with the leaves of the foliation F.
We use the normal exponential map to define a normal product neighborhood of a subset K ⊂ L for a leaf L. Given 0 < ≤ 0 , let Q(K, ) → K denote the restriction of the -disk bundle Q → M to K. The normal neighborhood N (K, ) is the image of the map, exp Q : Q(K, ) → M . If K = {x} is a point and 0 < < 0 , then N (x, ) is a uniformly transverse normal disk to F.
The restriction of the ambient metric d M to a leaf L need not coincide (locally) with the leafwise geodesic metric d F -unless the leaves of F are totally geodesic submanifolds of M . In any case, the Gauss Lemma implies that the two metrics are locally equivalent. We require that 0 > 0 satisfy: (5) For all x ∈ M , and for all y, y ∈ B F (x, 0 ), then d F and d M are related by Let dvol denote the leafwise volume p-form associated to the Riemannian metric on T F. Given any bounded, Borel subset A ⊂ L for the leafwise metric, define its leafwise volume by vol(A) = A dvol.
Let L ⊂ M be a compact leaf, then there exists 0 < L < 0 such that the normal geodesic map exp Q : Q(K, L ) → M is a diffeomorphism onto the open neighborhood N (K, L ). We thus obtain a normal projection map along the normal geodesic balls to points in L, which we denote by Π L : N (K, L ) → L. Note that the restriction of Π L to L is the identity map.
Let F|N (K, L ) denote the restricted foliation whose leaves are the connected components of the leaves of F intersected with N (K, L ). The tangent bundle to F|N (K, L ) is just the restriction of T F to N (K, L ), so for x ∈ N (K, L ) and x = Π L (x ), the differential of Π L induces a linear isomorphism D F Π L : T x F → T x L. Then the assumption on 0 in 2.1(4) above, implies that D F Π L satisfies a Lipschitz estimate for some constant C, which is the identity when restricted to the the leaf tangent bundle T L.
We use this observation in two ways. For L ⊂ M a compact leaf, assume that 0 < L ≤ 0 satisfies, for x ∈ L and x ∈ N (K, L ) such that x = Π L (x ): (7) for the leafwise Riemannian norm · F and v ∈ T x F, 2.2. Regular Foliation Atlas. We next recall some basic properties of foliation charts. A regular foliation atlas for F is a finite collection {(U α , φ α ) | α ∈ A} so that: (F4) P α (z) and P α (z) are convex subsets of diameter less than 1 with respect to d F .
The construction of regular coverings is described in chapter 1.2 of [3].
If the tangent bundle T F and normal bundle Q = T F ⊥ to F are oriented, then we assume that the charts in the regular covering preserve these orientations.
The inverse images P α (z) = φ −1 α ((−1, 1) p × {z}) ⊂ U α are smoothly embedded discs contained in the leaves of F, called the plaques associated to the given foliation atlas. The convexity hypotheses in (F4) implies that if U α ∩ U β = ∅, then each plaque P α (z) intersects at most one plaque of U β . The analogous statement holds for pairs U α ∩ U β = ∅. More generally, an intersection of plaques P α1 (z 1 ) ∩ · · · ∩ P α d (z d ) is either empty, or a convex set.
Recall that a Lebesgue number for the covering U is a constant > 0 so that for each x ∈ M there exists U ∈ U with B M (x, ) ⊂ U . Every covering of a compact Riemannian manifold (in fact, of a compact metric space) admits a Lebesgue number. We also require that 0 > 0 satisfy: (8) 2 0 is a Lebesgue number for the covering {U α | α ∈ A} of M by foliation charts.
Then for any x ∈ M , the restriction of F to B M (x, 0 ) is a product foliation, and by condition (F1) the leaves of F | B M (x, 0 ) are convex discs for the metric d F .
For each α ∈ A, the extended chart φ α defines a C 1 -embedding whose image is denoted by T α . We can assume that the images T α are pairwise disjoint. Let t α denote the restriction of t α to (−1, 1) q ⊂ (−2, 2) q , and define T α = t α (−1, 1) q . Then the collection of all plaques for the foliation atlas is indexed by the complete transversal For a point x ∈ T , let P α (x) = P α (t −1 α (x)) denote the plaque containing x. The Riemannian metric on M induces a Riemannian metric and corresponding distance function d T on each extended transversal T α . For α = β and x ∈ T α , y ∈ T β we set d T (x, y) = ∞.
Given a subset Z ⊂ U α let Z P denote the union of all plaques in U α having non-empty intersection with Z. We set Z T = Z P ∩ T α . If Z is an open subset of U α , then Z P is open in U α and Z T is an open subset of T α .
Then by the uniform extension property of the foliation charts, there exists a constant C T ≥ 1 so that for all α ∈ A, w ∈ (−1, 1) p and x, y ∈ T α (w), We use the maps (6) to translate points in the coordinate charts U α to the "center" transversal T α . The constant C T is a uniform estimate of the normal distortion introduced by this translation.
We will also consider the normal geodesic -disk N (y, ) at y = φ −1 α (w × 0), defined as the image of the map exp Q y : Q y → N (y, ), which for 0 < ≤ 0 is uniformly transverse to F. path-length distance function on N (y, ) for the restricted Riemannian metric on this transverse disk is equal to the distance function d M on M restricted to N (y, ).
Assume that the image N (y, ) ⊂ U α , then we can project it to the transversal T α along the plaques in U α . Denote this projection by Π F α : N (y, ) → T α . We also assume that the constant C T ≥ 1 is sufficiently large so that for all y ∈ M , for all 0 < ≤ 0 , for all α with N (y, ) ⊂ U α and for all z, z ∈ N (y, ) we have

Transverse holonomy.
The main result of this section is the definition of the module of uniform continuity function for elements of H n F , and its application in Lemma 2.1. We first recall the definition of the holonomy pseudogroup of F. A pair of indices (α, β) is said to be admissible if U α ∩ U β = ∅. Let T αβ ⊂ T α denote the open set of plaques of U α which intersect some plaque of U β . The holonomy transformation h αβ : T αβ → T βα is defined by y = h αβ (x) if and only if P α (x) ∩ P β (y) = ∅. The finite collection generates the holonomy pseudogroup H F of local homeomorphisms of T .
A plaque chain of length n, denoted by P, is a collection of plaques Each pair of indices (α i , α i+1 ) is admissible, so determines a holonomy map h αiαi+1 such that h αiαi+1 (z i ) = z i+1 . Let h P denote the composition of these maps, so that Let H n F = {h P | P has length at most n} ⊂ H F denote the collection of maps obtained from the composition of at most n maps in H 1 F . Each generator h αβ : T αβ → T βα is the restriction of the transition map h αβ : T αβ → T βα defined by the intersection of the extended charts U α ∩ U β . The domain T αβ ⊂ T αβ is precompact with B T (T αβ , 0 ) ⊂ T αβ , so h αβ is a uniformly continuous homeomorphism on its domain. That is, given any 0 < < 0 , there is a module of continuity µ αβ ( ) > 0 such that for all x ∈ T αβ For the admissible pairs (α, α) we set µ αα ( ) = . Given 0 < ≤ 0 , define (10) µ( ) = min{µ αβ ( ) | (α, β) admissible} so that 0 < µ( ) ≤ . Then for every admissible pair (α, β) and each x ∈ T αβ the holonomy map h αβ admits an extension to a local homeomorphism h αβ defined by the holonomy of F, which satisfies For an integer n > 0 and 0 < ≤ 0 recursively define µ (1) ( ) = µ( ) and µ (n) ( ) = µ(µ (n−1) ( )), so that µ (n) denotes the n-fold composition. Then define LEMMA 2.1. Given a plaque chain P of length n, and 0 < ≤ 0 set δ = µ(n, ). Then for any x in the domain of h P , there is an extension to a local homeomorphism h P defined by the holonomy of F whose domain includes the closure of the disk B T (x, δ) about x in T , and That is, µ(n, ) is a module of uniform continuity for all elements of H n F .
Proof. For each 0 ≤ i < n, µ(n, ) ≤ µ(i, ) hence there is an extension of is contained in a ball of radius at most µ(n − 1, ), so that we can continue the extension process to h i+1 .

Plaque length and metric geometry.
We make two observations about the metric leafwise geometry of foliations [26]. In particular, the technical result Proposition 2.3 below is a key fact for our proof of the main result of this work.
The plaque length of γ, denoted by ||γ|| P , is the least integer n such that the image of γ is covered by a chain of convex plaques Proof. Let N = (||γ|| F / 0 ) be the least integer greater than ||γ|| F / 0 . Then there exist points Thus, the image of γ is covered by a chain of convex plaques of length at most N .
The extension property (F2) in Section 2.2 implies that for all α ∈ A and z ∈ (−1, 1) q , the closure P α (z) is compact, hence has finite leafwise volume which is uniformly continuous with respect to the parameter z. Hence, there exist constants 0 < C min ≤ C max such that We note a consequence of this uniformity which is critical to the proof of the main theorem. Proof. The holonomy pseudogroup of F has a finite set of generators, hence has a uniform upper bound on the growth rate of words. This implies that given r > 0, there exists a positive integer e(r) such that any subset of a leaf with leaf diameter at most r can be covered by no more that e(r) plaques. Thus, if L is a leaf with diameter at most r, then L has volume at most v(r) = C max · e(r).
Now suppose that L is a compact leaf with diameter r = diam (L). Then, for any pair of points x, y ∈ L, there exists a length minimizing geodesic segment γ : (2). Recall that the restricted metric d F on leaves has uniformly bounded geometry. Then as 0 is assumed in assumption (2) of Section 2.1 to be less than the injectivity radius for the leafwise metric, and γ is a length-minimizing geodesic, there is a constant V 0 > 0 so that the leafwise volume vol (B F (γ, 0 )) ≥ V 0 · r. Thus, vol(L) > V 0 · diam (L), and then set R(v) = v/V 0 .

Captured leaves.
The main result is Proposition 2.6, which shows that given a compact leaf L of F, and another compact leaf L which is sufficiently close to L at some point, where how close depends on vol(L ), then L is "captured" by the holonomy of L. We require some preliminary definitions and observations before giving the proof of this key fact.
Let L ⊂ M be a compact leaf, and recall that in Section 2.1 the constant 0 < L ≤ 0 was defined so that there is a projection map Π L : N (L, L ) → L along the transverse geodesic L -disks to L.
Next, recall that Proposition 2.2 shows that for any leafwise C 1 -path γ, we have the upper bound ||γ|| P ≤ (||γ|| F / 0 ) for the number of plaques required to cover γ.
Suppose that L is a compact leaf, and x ∈ L is a fixed basepoint, then for any y ∈ L there is a leafwise geodesic γ x,y : [0, 1] → L from x to y with γ F ≤ diam(L). Thus, γ x,y can be covered by at most (||γ|| F / 0 ) plaques.
Recall that for n > 0, the number µ(n, 0 ) ≤ 0 was defined in (10), and the constant C T ≥ 1 was introduced in (7) and (8) as a bound on the distortion of the projection maps Π F α0 : N (x, 0 ) → T α0 . Introduce the function DEFINITION 2.4. For 0 < ≤ 0 and r > 0, (14) ∆ We scale both the domain variable and the range value of µ by C T so that we have uniform estimates for pairs of points in any geodesic normal ball in the chart, a fact which will be used later.
Recall that the constant C T was chosen so that for the projection Π F α0 : N (x, 0 ) → T α0 , and for x ∈ N (x, 0 ), the condition (8) implies that Given x ∈ N (x, ∆(r, )) then by (15), Then by Lemma 2.1 and the inclusion (12), we have that d T (h P (x), h P (x )) ≤ /C T .
We apply Lemma 2.5 to obtain the following fundamental property of compact foliations.
If P is the plaque chain containing γ x,z chosen in the proof of Lemma 2.5, then there is a corresponding plaque chain P = {P α0 (z 0 ), P α1 (z 1 ), . . . , P αn (z n )} in L from w to w , where each z i ∈ T αi . We say that P shadows in L the plaque chain P in L. In particular, we have w ∈ P αn (z n ) ⊂ L .
The conclusion of Lemma 2.5 is that for 2δ Λ = ∆(R(Λ), L ) then w ∈ N (z, L ), and hence d M (z, w ) < L . Thus, z ∈ N (L , L ) for all z ∈ L, hence L ⊂ N (L , L ). Proposition 2.6 has the following useful consequence.
By the assumption (4) in Section 2.1, the leaves of F are uniformly transverse to the fibers of Π L0 : N (L 0 , L0 ) → L 0 , so the restriction to L 1 is a covering map. As L 0 and L 1 are compact, the map Π L0 : L 1 → L 0 is onto. Assume that the tangent bundle T F is oriented, then we can choose a positively-oriented Riemannian volume form on the leaves of F, whose restriction to a leaf L is denoted by ω L . Then the closed p-form vol(L) −1 · ω L is dual to the fundamental class [L] of L.
The homological degree d * of a covering map equals its covering degree, and is given by Condition (6) of Section 2.1 gives that for x ∈ L 1 and x = Π L0 (x ) ∈ L 0 we have and thus combining (17) and (18) we obtain Thus, vol(L 1 ) ≤ 2d * · vol(L 0 ), as was to be shown.

Properties of compact foliations
In this section, F is assumed to be a compact foliation of a manifold M without boundary. The geometry of compact foliations has been studied by Epstein [11,12], Millett [22], Vogt [33,34,35] and Edwards, Millett and Sullivan [10]. We recall some of their results. The bad set X 1 of F consists of the points y ∈ M where the function v(x) is not bounded in any open neighborhood of y. By its definition, the bad set X 1 is saturated. Note also that The leaves in the intersection X 1 ∩ vol −1 (0, n] have volume at most n, while v(x) is not locally bounded in any open neighborhood of y ∈ X 1 , therefore each set X 1 ∩ vol −1 (0, n] has no interior. By the Baire category theorem, X 1 has no interior.
The complement G = M X 1 is called the good set. The holonomy of every leaf L ⊂ G is finite, thus by the Reeb Stability Theorem, L has an open saturated neighborhood consisting of leaves with finite holonomy. Hence, G is an open set, X 1 is closed, and the leaf space G/F is Hausdorff.
Inside the good set is the open dense saturated subset G e ⊂ G consisting of leaves without holonomy. Its complement G h = G G e consists of leaves with non-trivial finite holonomy.
3.2. The Epstein filtration. The restriction of the volume function v(x) to X 1 again need not be locally bounded, and the construction of the bad set can be iterated to obtain the Epstein filtration: The definition of the sets X α proceeds inductively: Let α > 1 be a countable ordinal, and assume that X β has been defined for β < α. If α is a successor ordinal, let α = γ + 1 and define X α to be the closed saturated set of y ∈ X γ where the function v(x) is not bounded in any relatively open neighborhood of y ∈ X γ in X γ .
For β < α, the set X α is nowhere dense in X β . Note that since each set M X α is open, the filtration is at most countable. The filtration length of F is the ordinal α such that X α = ∅ and X α+1 = ∅.
Vogt [35] showed that for any finite ordinal α, there is a compact foliation of a compact manifold with filtration length α. He also remarked that given any countable ordinal α, the construction can be modified to produce a foliation with filtration length α. Such examples show that the bad set X 1 and the subspaces X α need not be finite unions or intersections of submanifolds; they may have pathological topological structure, especially when the filtration length is an infinite ordinal.

Regular points.
A point x ∈ X 1 is called a regular point if the restricted holonomy of F|X 1 is trivial at x. Equivalently, the regular points are the points of continuity for the restricted volume function v|X 1 : It is precisely this "non-localized limit behavior" for leaves with unbounded volumes approaching the bad set which makes the study of compact foliations with non-empty bad sets so interesting, and difficult. There seem to be no results in the literature describing how these paths of leaves must behave in the limit.
3.4. Structure of the good set. Epstein [12] and Millett [22] showed that for a compact foliation F of a manifold V , then v(x) is locally bounded ⇔ V /F is Hausdorff ⇔ the holonomy of every leaf is finite .
By definition, the leaf volume function is locally bounded on the good set G, hence the restriction of F to G is compact Hausdorff, and all leaves of F|G have finite holonomy group. Epstein and Millett showed there is a much more precise structure theorem for the foliation F in an open neighborhood of a leaf of the good set: (1) there is a finite subgroup H x of the orthogonal group O(q) and a free action α x of H x on L 0 (2) there exists a diffeomorphism of the twisted product (20) φ In particular, if x ∈ V e then H x is trivial, and φ x is a product structure for a neighborhood of L x .
The open set V x is called a standard neighborhood of L x , and the 4- The Hausdorff space G/F is a Satake manifold; that is, for each point b ∈ G/F and π(x) = b the leaf L x has an open foliated neighborhood V x as above, and φ x : . The open sets W b ⊂ G/F are called basic open sets for G/F. Note also that π is a closed map [12,22].

Properties of foliated homotopies
In this section, we study some of the geometric and topological properties of a foliated homotopy of a compact leaf. These results play an essential role in our proof of Proposition 6.3, and hence of Theorem 1.2. The main result of this section is the following, which yields an upper bound on both the volumes of the compact leaves, and the topological degrees of the covering maps, which arise in a homotopy of a compact leaf. Note that the results of this section apply to any C 1 -foliation of a manifold M . Assume that both the tangent bundle T F and the normal bundle Q to F are oriented. Then there exists d * > 0, depending on H and L, such that Moreover, there exists an integer k ≥ 0 such that Proof. We first recall a result which holds for all C 1 -foliations. The homological degree of a covering map equals its covering degree, thus the covering degree of Π Lt : L s → L t divides the homological degree of deg(H t : L 0 → L t ).

The collection of open intervals {I
is an open covering of [0, 1], so there exists a finite set {0 = t 0 < t 1 < · · · < t k−1 < t k = 1} so that the collection {I ti | 0 ≤ i ≤ k} is a finite covering of [0, 1]. Choose a sequence {0 < s 1 < · · · < s k−1 < 1} such that and hence s ∈ I t −1 ∩ I t . Thus for the choices of the constants δ t for each 0 < < k, we have the inclusions L s ⊂ N (L t −1 , t −1 ) ∩ N (L t , t ). Thus, there are finite covering maps The collection of maps (24) is called a geometric correspondence from L 0 to L 1 . We have shown: Introduce the following integer constants associated to a correspondence (24), for 0 < < k: Note that a and b are equal to the covering degrees of the covering maps in (25) and (26), and that a 1 = 1 as the leaf L s1 must be a diffeomorphic covering of L 0 . Then the choice of each t ≤ 0 we can apply the estimate (4) as in the proof of Corollary 2.7 to obtain, for 1 ≤ < k, We can then combine these sequences of upper and lower estimates to obtain the estimate: and we obtain the estimate (22) for t = 1. The uniform bound (21) follows from the argument above, considering only the homological degrees of the covering maps and ignoring the volume estimates. A minor modification of the above arguments also yields these estimates for the values 0 < t < 1. This completes the proof of Proposition 4.1.

Tame points in the bad set
In this section, we introduce the concept of a "tame point" in the bad set X 1 , which is a point x ∈ X 1 that can be approached by a path in the good set. The main result of this section proves the existence of tame points, using a more careful analysis of the ideas of the Moving Leaf Proposition 3.1. Tame points are used in section 6 for studying the deformations of the bad set under foliated homotopy.
Recall that the bad set X 1 consists of the points y ∈ M where the leaf volume function v(x) is not bounded in any open neighborhood of y, and is closed, saturated and has no interior. A point x 1 ∈ X 1 is regular if the restriction of the leaf volume function v : X 1 → R + is continuous at x 1 .
Equivalently, x 1 ∈ X 1 is a regular point if the holonomy of the restriction of F to X 1 is trivial in some relatively open neighborhood of x 1 ∈ X 1 . with γ(t) ∈ G e for 0 ≤ t < 1, γ(1) = x 1 and such that v(γ(t))) tends uniformly to infinity as t → 1.
Let X t 1 ⊂ X 1 denote the subset of tame points. Since the restricted path γ : [0, 1) → G e lies in the set of leaves without holonomy, it follows that there is a foliated isotopy Γ : L γ(0) × [0, 1) → G e such that Γ t (γ(0)) = γ(t). Thus, a tame point x is directly approachable by a family of moving leaves whose volumes tend uniformly to infinity.
In the examples constructed by Sullivan [30], it is easy to see that every regular point is a tame point. In general, though, Edwards, Millet, and Sullivan specifically point out that their proof of the Moving Leaf Proposition 3.1 in [10] does not claim that a regular point is a tame point. The problem is due to the possibility that the complement of the bad set need not be locally connected in a neighborhood of a point in the bad set. In their proof, the moving leaf is defined by a curve that follows "an end of the good set" out to infinity, passing through points where the volume is tending to infinity along the way. This end of the good set is contained in arbitrarily small -neighborhoods of the bad set, but they do not control the behavior of the end. Thus, the existence of a tame point is asserting the existence of a "tame end" of the good set on which the volume function is unbounded, and which is defined by open neighborhoods of some point in the bad set.
PROPOSITION 5.2. Let F be a compact, C 1 -foliation of a manifold M , and assume that the tangent bundle T F and the normal bundle Q to F are oriented. Then the set of tame points X t 1 is dense in X 1 .
Proof. The proof of this result involves several technical steps, so we first give an overview of the strategy of the proof. Let x 1 ∈ X 1 be a regular point, and L 1 the leaf through x 1 . We use a key result in the proof of the Moving Leaf Lemma to obtain an open neighborhood U of x 1 in its transversal space, on which the volume function is unbounded. We then choose a regular point x * ∈ U ∩ X 1 which is sufficiently close to x 1 , so that the leaf L * through x * is a diffeomorphic covering of the leaf L 1 . Moreover, the point x * is approachable by a path in the good set. Then we argue by contradiction, that if the leaf volume function does not tend uniformly to infinity along this path, then each leaf through a point in the set U ∩ G is also a covering space of L 1 with uniformly bounded covering degree, from which we conclude that the volume function is bounded on the leaves through points in U , contrary to choice. It follows that x * is a tame point which is arbitrarily close to x 1 . The precise proofs of these claims requires that we first establish some technical properties of the foliation F in a normal neighborhood of L 1 .
Recall that in Section 2.1, given a compact leaf L the constant 0 < L ≤ 0 was defined so that there is a projection map Π L : N (L, L ) → L along the transverse geodesic L -disks to L. Set 1 = L1 so that the normal projection map Π L1 : In particular, the holonomy map h i along each closed path τ i is defined on the transverse disk N (x 1 , 2 ). That is, the transverse holonomy along τ i is represented by a local homeomorphism into The assumption that x 1 ∈ X 1 is a regular point implies that the germinal holonomy at x 1 of the restricted foliation F|X 1 is trivial. Thus we can choose 0 < 2δ ≤ 2 sufficiently small, so that each holonomy map h i restricted to X 1 ∩ N (x 1 , 2δ) is the identity map. It follows that the holonomy of F restricted to the closure (32) 2δ) is trivial. Hence, every point in Z 1 is a regular point of the bad set. It follows that the saturation Z F of Z 1 is a fibration over the closed set Z 1 , and that the leaf volume function v(y) is uniformly continuous and hence bounded on the compact set Z 1 . Thus, we may assume that δ is sufficiently small so that Z F ⊂ N (L 1 , 1 ). That is, for each z ∈ Z 1 the leaf L z ⊂ N (L 1 , 1 ).
Next consider the properties of the the normal projection Π L1 : N (L 1 , 1 ) → L 1 when restricted to leaves in N (L 1 , 1 ). The restriction π z ≡ Π|L z : L z → L 1 is a covering projection, which is a diffeomorphism as F|Z 1 has no holonomy, and by the assumption that 1 ≤ 0 , the estimate (5) implies the map π z is a quasi-isometry with expansion constant bounded by 2.
Note that N (x 1 , 2δ) is contained in the the normal transversal N (x 1 , 1 ), so by definition for z ∈ Z 1 we have π z (z) = x 1 , and thus given a path σ : For an arbitrary point y 0 ∈ N (x 1 , δ) and path σ : [0, 1] → L 1 with σ(0) = x 1 and path length σ ≤ D L1 , by the choice of δ the transverse holonomy map in (30) is defined at y 0 hence there is a lift of the path σ to a path σ y : [0, 1] → L y ∩ N (L 1 , 1 ) with σ y (0) = y 0 and π y • σ y (t) = σ(t) for all 0 ≤ t ≤ 1. This lifting property need not hold for paths longer than D L1 , as there may be leaves of F which intersect the normal neighborhood N (L 1 , δ) but are not contained in N (L 1 , 1 ).
We observe a technical point about the distances in the submanifold N (x 1 , 1 ) ⊂ M . The inclusion N (x 1 , 1 ) ⊂ M induces a Riemannian metric on N (x 1 , 1 ) which then defines a path-length distance function on this subspace. Unless N (x 1 , 1 ) is a totally geodesic submanifold of M , the induced distance function on N (x 1 , 1 ) need not agree with the restricted path-length metric from M . For y ∈ N (x 1 , 1 ) and 0 < λ ≤ 1 , let B T (y, λ) ⊂ N (x 1 , 1 ) denote the open ball of radius λ about y for the induced Riemannian metric on N (x 1 , 1 ). Now consider an arbitrary point y ∈ N (x 1 , δ) and assume that L y ⊂ N (L 1 , δ), so that L y is in the domain of the projection Π L1 : N (L 1 , 1 ) → L 1 . Given a path σ : [0, 1] → L y with σ(0) = y, then Lemma 2.5 implies that there exists 0 < λ < δ, which depends on the length σ , so that for y ∈ B T (y, λ) there is a path σ y : [0, 1] → L y satisfying Π Ly (σ(t)) = Π Ly (σ y (t)) for 0 ≤ t ≤ 1. We call the path σ y a lifting of σ y from L y to L y .
After these technical preliminary results, we begin the proof of Proposition 5.2. First, recall a key fact from the proof of the Moving Leaf Proposition 3.1, whose proof was in turn based on ideas of Montgomery [24] and Newman [25]. (In particular, Figure 3 on page 23 of [10] and the arguments following it are pertinent to the arguments below.) Next, fix a choice of regular point x 1 ∈ X 1 and sufficiently small constant δ > 0 as above so that (32) holds, then choose a point y 0 ∈ U ∩ N (x 1 , δ/2). Let x * ∈ Z 1 be a closest point to y 0 for the induced metric on N (x 1 , δ). That is, consider the sequence of closed balls B T (y 0 , λ) ⊂ N (x 1 , δ) Z 1 for λ > 0, expanding until there is a first contact with the frontier of U , then x * is contained in this intersection. Let δ 0 ≤ δ/2 denote the radius of first contact, hence δ 0 equals the distance from y 0 to x * in the induced path-length metric on N (x 1 , 1 ). Then B T (y 0 , δ 0 ) ⊂ U and x * ∈ B T (y 0 , δ 0 ) ∩ Z 1 . (This is illustrated in Figure 1 below.) Let L * = L x * denote the leaf containing x * .
We claim that x * is a tame point. As δ > 0 was chosen to be arbitrarily small, and the regular points are dense in the bad set, the proof of Proposition 5.2 then follows from this claim.
By the choice of B T (y 0 , δ 0 ) ⊂ U , there is a path γ : [0, 1] → N (x 1 , δ) such that γ(0) = y 0 , γ(1) = x * and γ[0, 1) ⊂ B T (y 0 , δ 0 ). The complement of X 1 is the good set, hence the image γ[0, 1) ⊂ G. The set of leaves with holonomy G h in G is a union of submanifolds with codimension at least 2 by Proposition 3.2. Thus, by a small C 1 -perturbation of the path γ in U , we can assume that its image is disjoint from the set G h . That is, γ(t) ∈ G e for all 0 ≤ t < 1, and γ(1) ∈ L * . Let L t denote the leaf containing γ(t).
We claim that the volumes of the leaves L t tend uniformly to infinity. Assume not, so there exists a constant V max > 0 and a sequence 0 < t 1 < · · · < t n · · · → 1 such that x n = γ(t n ) → x * and the volumes of the leaves L n = L xn are bounded above by V max . We show this yields a contradiction to our assumptions. What we show in the following is that if there exists a leaf L y for y ∈ U ∩ G e sufficiently close to L * with prescribed bounded volume, then using Proposition 2.6 and Corollary 2.7, we show this implies that all leaves intersecting U have bounded volume, which yields the contradiction. PROPOSITION 5.4. For V max > 0, there is an * > 0 so that if there exists y ∈ U ∩ G e such that d(y, x * ) < * and vol(L y ) ≤ V max , then for all y ∈ U , the leaf L y containing y has the volume bound vol(L y ) ≤ 2 V max .

Proof. By Proposition 2.3, there is a function
Recall that δ was chosen so that 2δ ≤ 2 where 2 = ∆(D L1 , 1 ) was defined after (29), and so that (32) holds, hence Z F ⊂ N (L 1 , 1 ). Thus by the choice x * ∈ B T (y 0 , δ 0 ) ∩ Z 1 , we have that L * = L x * ⊂ N (L 1 , 1 ). Let π * = π x * : L * → L 1 denote the normal projection, whose restriction to L * is a covering map, which is a diffeomorphism as x * ∈ Z 1 . We next choose y ∈ U ∩G e which is sufficiently close to L * so that L y ⊂ N (L 1 , 1 ) and the holonomy maps of L * based at x * are defined on L y . This will imply that L y is a finite covering of L * . , . . . , [τ * k ]} form a generating set for π 1 (L * , x * ). Denote the holonomy along τ * i by h * i . As L * ⊂ N (L 1 , 1 ), there exists 0 < 3 ≤ 2 be such that N (L * , 3 ) ⊂ N (L 1 , 1 ).
As L y is a compact leaf, its intersection with the transversal N (x * , 3 ) is a finite set, denoted by Then for each 1 ≤ i ≤ k, by (34) the holonomy map h * i satisfies h * i (F y ) ⊂ L y ∩ N (x * , 3 ) = F y . Thus, the finite set of points F y is permuted by the action of a set of generators for π 1 (L * , x * ). Thus, compositions of the generators are defined on the set F y . That is, for any w ∈ π 1 (L * , x * ) the holonomy h * w along w contains the finite set F y in its domain. Let H * ⊂ π 1 (L * , x * ) denote the normal subgroup of finite index consisting of all words whose holonomy fixes every point in F y .
Let z ∈ F y . For each w ∈ H * , the holonomy h * w map is defined at z, and so must be defined on some open neighborhood z ∈ V w z ⊂ U of z, where the diameter of the set V w z depends on z and w. As y ∈ U ∩ G e the leaf L y ⊂ G e is without holonomy, so the restriction of h * w to the open set V w z must fix an open neighborhood in N (x * , 1 ) of z ∈ U w z ⊂ V w z . Thus, the fix-point set of h * w contains an open neighborhood of F y in N (x * , 1 ). Since y ∈ L y ∩ N (x * , 3 ) = F y , we have in particular that there is an open neighborhood y ∈ U w y ⊂ U ∩ B T (x * , * ) contained in the fixed-point set for h * w . We next use these conclusions for the holonomy of the leaf L * to deduce properties of the holonomy for the leaf L 1 . Recall that N (L * , 3 ) ⊂ N (L 1 , 1 ), and each path τ * i in L * is the lift of the path τ i in L 1 via the covering map π * ≡ Π|L * : L * → L 1 . Thus, the holonomy map h Then the closed path representing w in L * can be lifted to a path τ y w in the leaf L y , and as L y ⊂ N (L 1 , 1 ), its length is bounded above by τ y w ≤ m * · 2D L1 . We show that h w is defined on U , and U ⊂ Fix(h w ). This implies that there is a uniform bound on the diameter of the leaves L y for y ∈ U , from which it follows that there is an upper bound on the function vol(L y ) for y ∈ U , which contradicts the choice of U .
We first show that L y is a finite covering of L 1 with the same index as the covering L y → L 1 .

Figure 2. A path chain in the good set
We show that σ([0, 1]) ⊂ Fix(h w ) using induction on the index i. For i = 0, y 0 = y and by assumption, the disk B T (y, δ * ) ⊂ U w y ⊂ Fix(h w ) so σ([0, t 1 ]) ⊂ Fix(h w ). Now assume σ([0, t n ]) ⊂ Fix(h w ), hence y n = σ(t n ) ∈ Fix(h w ). The closed path τ * w in L * representing w is the lift of a closed path τ w in L 1 , which lifts to a closed path τ yn w in L yn . As τ yn w ⊂ N (L 1 , 1 ) we have that τ yn w ≤ 2m * D L1 . Then the holonomy map h w for w fixes y n so near y n it is defined by a map h yn w : N (y n , δ * ) → N (y n , * ) .
As the points of U ∩ G e determine leaves without holonomy, the set of fixed-points for h yn w is an open subset of N (y n , δ * ) ∩ U ∩ G e . The set of fixed-points is also always a (relatively) closed subset, hence Fix(h yn w ) contains the connected component of N (y n , δ * ) ∩ U ∩ G e which contains the point y n . By assumption we have that σ([t n , t n+1 ]) ⊂ N (y n , δ * ) ∩ U ∩ G e , hence (36) σ . Thus, by induction we conclude that y ∈ Fix(h w ).
The choice of y ∈ U was arbitrary, and thus U ⊂ Fix(h w ). We conclude that L y is a finite covering of L 1 and isotopic to L y , hence vol(L y ) ≤ 2 V max . This completes the proof of Proposition 5.4.
The proof of Proposition 5.2 then follows, as the points in the curve σ(t) have limit x * as t → 1.

Proof of Main Theorem
In this section, we show that for a compact foliation of a compact manifold, a categorical open set cannot contain a tame point. A categorical set must be connected, so we may assume that M is connected. For a connected manifold M , there is a finite covering (of degree d ≤ 4) M → M for which the lifted foliation F is again compact, and has oriented tangent and normal bundles. We then apply the following two elementary results to reduce to the oriented case. Proof. The covering map π has the unique local lifting of paths property, so in particular has the homotopy lifting property, which yields the existence of the lifted homotopy H. LEMMA 6.2. Let π : M → M be a finite covering of degree 1 < d < ∞, with foliation F whose leaves are finite coverings of the leaves of F. Let L ⊂ M be a leaf of F, and let x ∈ L with x = π( x). Then x is a tame point in the bad set for F, if and only if x is a tame point in the bad set for F.

Proof.
Suppose that x is a tame point for F, then for > 0 there exists a continuous path γ : [0, 1] → M with γ(1) = x, as in Definition 5.1. The map π has the unique local lifting of paths property, so there exists a unique path γ : [0, 1] → M with γ(1) = x. Moreover, for each 0 ≤ t < 1 the leaf L t containing γ(t) is a finite covering of the leaf L t ⊂ M containing γ(t) where the degree π : L t → L t has degree at most d. Thus, the volume vol(L t ) tends to infinity as t → 1, and thus the same holds for the volume function vol( L t ) in M . Thus, x is a tame point in the bad set for F. Conversely, if x is tame point for F then the proof that x is a tame point for F follows similarly.
Here is the main result of this section. Proof. As a consequence of Lemma 6.2, we can assume in the following that both the tangent bundle T F and the normal bundle Q to F are oriented. Let x 1 ∈ X 1 be a tame point, V ⊂ M an open set with x 1 ∈ V , and suppose there exists a leafwise homotopy H : V × [0, 1] → M with H 0 = Id, and H 1 (V ) ⊂ L * for some leaf L * . We show that this yields a contradiction.
Recall that for x ∈ M , we let v(x) denote the volume of the leaf L x containing x.
As x 1 is a tame point, there is a smooth path γ : [0, 1] → V such that γ(1) = x 1 , γ(t) ∈ G e for 0 ≤ t < 1, and the volume v(γ(t)) of the leaf L t containing the point γ(t) satisfies lim  (γ(t)). The key to obtaining a contradiction is to analyze the behavior of the leaf volume function v(φ(s, t)).
. Then x t ∈ G e for 0 < t ≤ 1, while x 1 ∈ X 1 is the given tame point.
Also note that for t = 0, and each 0 ≤ s ≤ 1, the map H s : L 0 = L 0,0 → L s,0 is surjective by The set V is saturated, so for each 0 ≤ t < 1, the leaf L t ⊂ V as γ(t) ∈ V . Thus, we can define a continuous 2-parameter family of maps Φ : [0, 1] × [0, 1) × L 0 → M by setting Φ s,t (y) = H s (Γ t (y)) for y ∈ L 0 . It is important to recall the usual caution with the study of compact foliations: the path of leaves t → L t with unbounded volumes cannot limit on a compact leaf in the bad set. Thus, the paths s → L s,t must become more chaotic as t → 1, and correspondingly, the family of maps Φ s,t is not defined for t = 1. On the other hand, we are given that the path γ(t) limits on x 1 and so the trace Φ s,t (x 0 ) extends to the continuous map φ(s, t) = H s (γ(t)) for t = 1. We use this extension of Φ s,t (y) for y = x 0 to show that the map extends for all y ∈ L 0 which gives a contradiction.
The idea of the proof of the existence of this extension is to use the techniques for studying a homotopy of compact leaves introduced in Section 4, to control the degrees of the maps on the fundamental classes of the leaves, induced by the maps Φ s,t . This will in turn yield bounds on the volumes of these leaves, which yields bounds on their diameters. We can thus use Proposition 2.6 to conclude that for t * < 1 sufficiently close to t = 1, for each 0 ≤ s ≤ 1, the image Φ s,t * (L 0 ) is contained in a uniform normal neighborhood of H s (L 1 ), from which the claim follows. The details required to fill out this sketch of proof are tedious, but otherwise straightforward.
First, observe that Φ 1,t : L 0 → L * , for 0 ≤ t < 1, is a family of homotopic maps, hence its homological degree is constant. Thus, for all 0 ≤ t < 1 we have: It follows that Let D = R(2d 0 d 1 D L ) be the maximum diameter of a leaf with volume at most 2 d 0 d 1 D L , where we recall that D L is defined in (37) and d 0 and d 1 are defined in the text immediately preceding (37).
Let L be a compact leaf such that vol(L) ≤ 2 d 0 d 1 D L and L ∩ N (L s,1 , δ s ) = ∅, then by the choice of D and δ s , Proposition 2.6 implies that L ⊂ N (L s,1 , s ). Thus, the restriction Π Ls,1 : L → L s,1 is well-defined and a covering map, and moreover by Corollary 2.7 we have the estimate Choose a sequence 0 = s 0 < s 1 < · · · < s N −1 < s N = 1 of points such that for λ n = λ Set δ n = δ sn and n = sn for 0 ≤ n ≤ N , and λ * = min{λ n | n = 0, 1, . . . , N } > 0.
Here is the key result: Proof. For each 1 ≤ n ≤ N , set ξ 0 = 0 and ξ N +1 = 1, and for 1 ≤ n ≤ N choose points ξ n ∈ (s n−1 , s n−1 + λ n−1 ) ∩ (s n − λ n , s n ) . Let µ satisfy 1 − λ * ≤ µ < 1, and let L µ = Γ µ (L 0 ) be the leaf through γ(µ). The technical idea of the proof of (41) is to compare the homological degrees of the maps using a downward induction argument on n, starting with n = N , and showing there is a uniform bound on the ratios of their degrees for all 1 − λ * ≤ µ < 1.
Note that L 1,µ = L 1,1 = L * , thus for s < 1 sufficiently close to 1 we have H s (L µ ) ⊂ N (L 1,1 , N ) as the homotopy H s is uniformly continuous when restricted to the compact leaf L µ .
Let r N be the infimum of r such that r ≤ s ≤ 1 implies L s,µ ⊂ N (L 1,1 , N ). The above remark implies r N < 1. We claim that r N < ξ N .
As we assumed that r N ≥ ξ N ≥ λ N we have that φ(r N , µ) ∈ N (L 1,1 , δ N ), hence Proposition 2.6 implies that L r N ,µ ⊂ N (L 1,1 , N ). By the uniform continuity of H s restricted to L µ at s = r N , there is r < r N such that r < s ≤ r N implies L s,µ ⊂ N (L 1,1 , N ). This contradicts the choice of r N as the infimum of such r, hence we must have that r N < ξ N .
This proves the first statement of the inductive hypothesis for n = N , which is that the estimate (44) holds for all ξ N ≤ r ≤ 1 and 1 − λ * ≤ µ < 1.
We next consider the ratios of covering degrees for a pair of leaves in adjacent normal neighborhoods. For ξ N ≤ s ≤ 1, we have φ(s, 1) ∈ N (L 1,1 , δ N ) and vol(L s,1 ) ≤ D L hence L s,1 ⊂ N (L 1,1 , N −1 ), and so the normal projection restricts to a covering map Π L1,1 : L s,1 → L 1,1 . Moreover, this implies that both L ξ N ,µ and L ξ N ,1 are coverings of L 1,1 , and their homological degrees are denoted by Note that as s N −1 < ξ N , the leaves L ξ N ,µ and L ξ N ,1 are also coverings of L s N −1 ,1 . We compare their homological degrees. By the uniform continuity of H s restricted to the curve γ(t), for 0 ≤ s ≤ 1, the path t → φ(s, t) has limit x s,1 = H s (x 1 ). By Proposition 2.6, the volume bound (44) for 1 − ξ N ≤ s ≤ 1 and 1 − λ * ≤ t < 1 implies that Thus, there is a well-defined limit The terminology deg (Φ s,1 : L 0 → L s,1 ) is a small abuse of notation, as given y ∈ L 0 there is no assurance that t → Φ s,t (y) has a limit at t = 1; it is only given that the image is trapped in the open neighborhood N (L s,1 , s ), and the images are homotopic for t sufficiently close to 1.
We show that if (52) and (53) are true for n, then the corresponding statements are true for n − 1.
Thus, the restriction Π Ls n−1 ,1 : L ξn,t → L sn−1,1 is a covering map. The key to the proof of the inductive step is to obtain a uniform estimate for the homological degree of this covering map.
Proof. Consider the diagram  where the integer next to a covering map indicates its homological degree.
As deg H sn−1 : L 0,1 → L sn−1,1 = d s,1 ≤ d 1 and the degrees of the maps are positive integers, it follows that 1 ≤ b n ≤ d 1 .
The leaf volume function is lower semi-continuous, hence we also have that Thus, the estimate (64) holds for all r n−1 ≤ r ≤ 1 and 1 − λ * ≤ µ < 1.
The second inductive statement (53) follows exactly as before.
The conclusion (41) of Lemma 6.4 for s = 0 contradicts the assumption that lim t→∞ vol(L 0,t ) = ∞. This contradiction completes the proof of Proposition 6.3.