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Realization of manifolds as leaves using graph colorings

It is proved that any (repetitive) Riemannian manifold of bounded geometry can be realized as a leaf of some (minimal) Riemannian matchbox manifold without holonomy. Our methods can be adapted to achieve Cantor transversals or a prescribed holonomy covering, but then the manifold may not be realized as a dense leaf.

1. Introduction 1.1. Realization of manifolds as leaves. Sondow [47] and Sullivan [48] began the fundamental study of which connected manifolds can be realized as leaves of foliations on compact manifolds. A manifold is called a leaf or non-leaf if the answer is positive or negative, respectively. In codimension one, Cantwell and Conlon [16] have shown that any open connected surface is a leaf, whereas Ghys [23], Inaba et al. [30], and Schweitzer and Souza [44] constructed non-leaves of dimension 3 and higher. Other non-leaves in codimension one, with exotic differential structures, were constructed by Meniño Cotón and Schweitzer [35].
Any leaf of a foliation on a compact Riemannian manifold M is of bounded geometry, and its quasiisometry type is independent of the metric on the ambient manifold. Thus it is also natural to study which connected Riemannian manifolds of bounded geometry are quasi-isometric to leaves of foliations on compact manifolds. This metric version of the realization problem was studied by Phillips and Sullivan [38], Januszkiewicz [31], Cantwell and Conlon [13][14][15], Cass [17], Schweitzer [42,43], Attie and Hurder [9], and Zeghib [49], constructing examples of non-leaves in codimension one and higher.
This realization problem can be also considered using compact (Polish) foliated spaces. On foliated spaces, differentiable structures or Riemannian metrics refer to the leafwise direction, keeping continuity on the ambient space. Like in the case of foliations, any leaf of a compact Riemannian foliated space is of bounded geometry. The converse statement is also true, in contrast with the case of foliations on compact manifolds; actually, any connected Riemannian manifold of bounded geometry is isometric to a leaf without holonomy in some compact Riemannian foliated space [4, Theorem 1.1] (see also [6,Theorem 1.5]). Another interesting realization of hyperbolic surfaces as leaves of compact foliated spaces was achieved in [3].
Besides achieving realization in matchbox manifolds, Theorem 1.1 improves [4, Theorem 1.1] by removing holonomy from all leaves, and achieving minimality in the case of repetitive manifolds. Thus Theorem 1.1 implies the converse of the following implication: in any minimal compact Riemannian foliated space, all leaves without holonomy are repetitive (Proposition 2.22).
For example, Theorem 1.1 can be applied to any complete connected hyperbolic manifold with positive injectivity radius. It can be also applied to any connected Lie group with a left invariant metric. Some of them are not coarsely quasi-isometric to any finitely generated group [18,22], obtaining compact, minimal, Riemannian matchbox manifolds without holonomy whose leaves are isometric to each other, but not coarsely quasi-isometric to any finitely generated group.
Since any smooth C ∞ manifold admits a metric of bounded geometry [25], it follows from Theorem 1.1 that any C ∞ connected manifold can be realized as a leaf of a C ∞ matchbox manifold without holonomy. For instance, this is true for the exotic 4-manifolds that are non-leaves in codimension one [35].
In Theorem 1.1, the realization of leaves in smooth matchbox manifolds without holonomy is relevant because they are homeomorphic to a projective limit of maps between compact branched manifolds [2,20].
This was generalized to arbitrary matchbox manifolds in [33], but the proof has a gap, even though the result might be correct.
In the following consequences of Theorem 1.1, the realization of a Riemannian manifold as a leaf is achieved with some additional properties, but losing the density of that leaf. Corollary 1.2. Any non-compact connected Riemannian manifold of bounded geometry is isometric to a leaf in some Riemannian matchbox manifold without holonomy that has a complete transversal homeomorphic to a Cantor space.
The Levi-Civita connection determines a decomposition T (2) M = H ⊕ V, as direct sum of the horizontal and vertical subbundles. Consider the Sasaki metric g (1) on T M , which is the unique Riemannian metric such that H ⊥ V and the canonical identities H ξ ≡ T ξ M ≡ V ξ are isometries for every ξ ∈ T M . For m ≥ 2, consider the Sasaki metric g (m) = (g (m−1) ) (1) on T (m) M . The notation d (m) is used for the corresponding distance function, and the corresponding open and closed balls of center v ∈ T (m) M and radius r > 0 are denoted by B (m) (v, r) and D (m) (v, r). For l < j, T (l) M is totally geodesic in T (m) M and g (m) | T (l) M = g (l) .
Let D ⊂ M be a compact domain 3 and m ∈ N. The C m tensors on D of a fixed type form a Banach space with the norm C m ,D,g defined by A C m ,D,g = max 0≤l≤m, x∈D |∇ l A(x)| .
By taking the projective limit as m → ∞, we get the Fréchet space of C ∞ tensors on D of that type equipped with the C ∞ topology (see e.g. [29]). Similar definitions apply to the space of C m or C ∞ functions on M with values in a separable Hilbert space (of finite or infinite dimension).
For m ∈ N, a partial map h : M M ′ is called a C m local diffeomorphism if dom h and im h are open in M and M ′ , respectively, and h : dom h → im h is a C m diffeomorphism. If moreover h(x) = x ′ for distinguished points, x ∈ dom h and x ′ ∈ im h, then h is said to be pointed, and the notation h : (M, x) (M ′ , x ′ ) is used. The term (pointed ) local homeomorphism is used in the C 0 case.
For m ∈ N, R > 0 and λ ≥ 1, an (m, R, λ)-pointed partial quasi-isometry 4 (or simply an (m, R, λ)-p.p.q.i.) is a pointed partial map h : (M, x) (M ′ , x ′ ), with dom h = D(x, R), which can be extended to a C m+1diffeomorphismh between open subsets such that D  (i) sup s∈S sup D |∇ k s| < ∞ for every compact subset D ⊂ M and 1 ≤ k ≤ m + 1; and (ii) { (∇ k s)(x 0 ) | s ∈ S } is precompact in 5 E x0 ⊗ k T * x0 M for all 0 ≤ k ≤ m. Proof. We proceed by induction on m. Consider the case m = 0. From (i) for k = 1, it follows that S is equicontinuous on the interior of D, and therefore on M because D is an arbitrary compact subset. Moreover (ii) for k = 0 states that { s(x 0 ) | s ∈ S } is precompact in E x0 . So S is precompact in C(M ; E) by the Arzelà-Ascoli theorem. Now assume that m ≥ 1 and the result is true for m − 1. Given x ∈ M , 0 ≤ t, u ≤ 1 and a piecewise smooth path c : [0, 1] → M from x 0 to x, let P u c,t : E c(t) → E c(u) be the ∇-parallel transport along c from u to v. For any e ∈ E x0 and α ∈ C m−1 (M ; E ⊗ T * M ), let Q c (e, α) = P 1 c,0 (e) This expression defines a continuous map Q c : E x0 × C m−1 (M ; E ⊗ T * M ) → E x . In particular, for any s ∈ C m (M ; E), we have Q c (s(x 0 ), ∇s) = s(x) (2.1) 3 A regular submanifold of the same dimension as M , possibly with boundary. 4 The extensionh is an (m, R, λ)-pointed local quasi-isometry, as defined in [4]. On the other hand, any (m, R, λ)-pointed local quasi-isometry defines an (m, R, λ)-pointed partial quasi-isometry by restriction. Thus both notions are equivalent. 5 Ex 0 ⊗ k T * x 0 M ≡ Hom( k Tx 0 M, Ex 0 ) is endowed with the topology of uniform convergence over bounded subsets, induced by the operator norm. It agrees with the topology of pointwise convergence because dim k Tx 0 M < ∞.
So ∇s = α, and therefore T s = (e, α). Thus (e, α) ∈ im T , completing the proof of (a). The above argument also shows that T is injective, and T −1 : im T → C m (M ; E) is given by T −1 (e, α)(x) = Q c (e, α), where c : [0, 1] → M is any piecewise smooth path from x 0 to x. Thus T −1 : im T → C m (M ; E) is continuous, showing that T is an embedding.
Finally, im T is closed by (a) and the continuity of Q d : E x0 × C m−1 (M ; E ⊗ T * M ) → E x0 for every d ∈ Ω(M, x 0 ). Thus T is also a closed map, and the proof of Claim 1 is finished.
where the first factor is already known to be precompact in E x0 . On the other hand, we have ∇(S) ⊂ C m (M ; E ⊗ T * M ), and this subspace satisfies Proof. The "only if" part follows from the continuity of the operators The "if" part is true by Proposition 2.8 since C ∞ (M ; E) = m C m (M ; E) with the inverse limit topology.
Recall that M is said to be of bounded geometry if inj M > 0 and sup M |∇ m R M | < ∞ for all m ∈ N. For a given manifold M of bounded geometry, the optimal bounds of the previous inequalities will be referred to as the geometric bounds of M . Let B r = B R n (0, r) (r > 0).  [21]). M is of bounded geometry if and only if there is some 0 < r 0 < inj M such that, for normal parametrizations κ x : B r0 → B M (x, r 0 ) (x ∈ M ), the corresponding metric coefficients, g ij and g ij , as a family of C ∞ functions on B r0 parametrized by x, i and j, lie in a bounded subset of the Fréchet space C ∞ (B r0 ). . Suppose that M is of bounded geometry. For every τ > 0, there is some map c : R + → N, depending only on τ and the geometric bounds of M , such that, for any τ -separated subset X ⊂ M , and all x ∈ M and δ > 0, we have |D(x, δ) ∩ X| ≤ c(δ). Proposition 2.12. Let X be a τ -separated η-relatively dense subset of a manifold of bounded geometry M for some 0 < τ < η. Given 0 < ε < τ /2 and σ > 0, let τ ′ = τ − 2ε and η ′ = η + ε. Then there is some 0 < P = P (ε) < σ, depending only on τ , ε, σ and the geometric bounds of M , such that P (ε) → 0 as ε → 0 and, for every 0 < ρ < P and In particular, X ′ is τ ′ -separated and η ′ -relatively dense.
By (a), the subset has cardinality at most C. Thus, by (c) and (b), for all x ∈ Y \ B, So there is somex ∈ B(x, ε) such thatx / ∈ C(y, σ − ρ, σ + ρ) for every y ∈ Z. Thereforex / ∈ C(y, σ − ρ, σ + ρ) for all y ∈ Y , and Claim 2 follows. Let x 1 , x 2 , . . . be a (finite or infinite) sequence enumerating the elements of X \ A. Then X ′ is defined as the union of A and a sequence of elements In particular, X ′ will be an ε-perturbation of X. Let us define x ′ i by induction on i as follows. We use the notation X 0 = X and Note that X i is also an ε-perturbation of X and therefore τ ′ -separated. Assume that X i−1 is defined for some i ≥ 1. By Claim 2, we can take some The resulting set X ′ satisfies the desired properties; in particular, it is a τ ′ -separated η ′ -relatively dense subset of M by Lemma 2.6. Proposition 2.13. Let X be an ε-relatively dense subset of M for some ε > 0, and let h be an isometry of M . If ε is small enough and h = id on X, then h = id on M .
2.4. Foliated spaces. A foliated space (or lamination) X ≡ (X, F) of dimension n is a Polish space X equipped with a partition F (a foliated or laminated structure) into injectively immersed manifolds (leaves) so that X has an open cover {U i } with homeomorphisms φ i : U i → B i × T i , for some open balls B i ⊂ R n and Polish spaces T i , such that the slices B i × { * } correspond to open sets in the leaves (plaques); every (U i , φ i ) is called a foliated chart and U = {U i , φ i } a foliated atlas. The corresponding changes of foliated coordinates are locally of the form φ i φ −1 j (y, z) = (f ij (y, z), h ij (z)). Let p i : U i → T i denote the projection defined by every φ i , whose fibers are the plaques. The subspaces transverse to the leaves are called transversals; for instance, the subspaces φ −1 i ({ * } × T i ) ≡ T i are local transversals. A transversal is said to be complete if it meets all leaves. X is called a matchbox manifold if it is compact and connected, and its local transversals are totally disconnected.
We can assume that U is regular in the sense that it is locally finite, every φ i can be extended to a foliated chart whose domain contains U i , and every plaque of U i meets at most one plaque of U j . In this case, the maps h ij define unique homeomorphisms h ij : This H is unique up to Haefliger's equivalences [27,28], and its equivalence class is called the holonomy pseudogroup.
The H-orbits are equipped with a connected graph structure so that a pair of points is joined by an edge if they correspond by some h ij . The projections p i define an identity between the leaf space X/F and the orbit space T/H. Moreover we can choose points y i ∈ B i so that the corresponding local transversals Then their union is a complete transversal homeomorphic to T, and the H-orbits are given by the intersection of the complete transversal with the leaves. If X is compact, then U is finite, and therefore the vertex degrees of the H-orbits is bounded by the finite number of maps h ij . Moreover the coarse quasi-isometry class of the H-orbits is independent of U in this case.
If the functions y → f ij (y, z) are C ∞ with partial derivatives of arbitrary order depending continuously on z, then U defines a C ∞ structure on X, and X becomes a C ∞ foliated space with such a structure. Then C ∞ bundles and their C ∞ sections also make sense on X, defined by requiring that their local descriptions are C ∞ in a similar sense. For instance, the tangent bundle T X (or T F) is the C ∞ vector bundle over X that consists of the vectors tangent to the leaves, and a Riemannian metric on X consists of Riemannian metrics on the leaves that define a C ∞ section on X. This gives rise to the concept of Riemannian foliated space. If X is a compact C ∞ foliated space, then the differentiable quasi-isometry type of every leaf is independent of the choice of the Riemannian metric on X, and is coarsely quasi-isometric to the corresponding H-orbits (see e.g. [8,Section 10.3]).
Many of the concepts and properties of foliated spaces are direct generalizations from foliations. Several results about foliations have obvious versions for foliated spaces, like the holonomy group and holonomy cover of the leaves, and the Reeb's local stability theorem. This can be seen in the following standard references about foliated spaces: [36], [11,Chapter 11] 8 . In other words, for all m ∈ N, R, ε > 0 and λ > 1, there is an (m, R, λ)-  In [4,6,8], the notation M * (n) and M * (n) was used instead of M n * and M n * , adding the superindex "∞" when equipped with the topology defined by the C ∞ convergence. 7 The cardinality of each complete connected Riemannian n-manifold is less than or equal to the cardinality of the continuum, and therefore it may be assumed that its underlying set is contained in R. With this assumption, M n * is a well defined set. 8 The C m+1 embeddings and C m convergence of [ The last part of the statement follows from the first part and Lemma 2.
It is said that (M, f ) (or f ) is (locally) non-periodic (or (locally) aperiodic) ifι M,f is (locally) injective; i.e., aperiodicity means Iso(M, f ) = {id M }, and local aperiodicity means that the canonical projection M → Iso(M, f )\M is a covering map. More strongly, On the other hand, (M, f ) (or f ) is said to be repetitive if, given any p ∈ M , for all m ∈ N, R, ε > 0 and λ > 1, the points x ∈ M such that form a relatively dense subset of M . Clearly, this property is independent of the choice of p.
Take some x ∈ X with d M (x, y) ≤ c. We have D M (y, c) ⊂ h(D M ′ (y ′ , λ 1/2 c)), and therefore there is some Since m, R, ε and λ are arbitrary, we get [M, To prove (ii), fix any p ∈ M , and take m ∈ N, R, ε > 0 and λ > 1. The set Like in the case of M n * , it can be proved that this convergence defines a Polish topology on T M n * . Moreover there are continuous maps Tι M,f : , whose images T[M, f ] form a canonical partition of T M n * satisfying the same basic properties as the canonical partition of M n * . We also have a continuous forgetful or underlying map u : The above definition can be modified in obvious ways, giving rise to other partitioned spaces with the same basic properties. For instance, by using cotangent spaces T * x M instead of the tangent spaces T x M , we get a partitioned space T * M n * , where the partition is defined by the images Next, for k ∈ N, we can also use the tensor products k T x M or k T * x M , giving rise to partitioned spaces k T M n * and k T * M n * . Also, we can only take vectors v in the disks D r T x M ⊂ T x M of center zero and radius r ≥ 0, producing a partitioned subspace Similarly, we get partitioned subspaces D r T * M n * , D r k T M n * and D r k T * M n * . A a continuous forgetful or underlying map u is defined in all of these spaces with values in M n * . We will use the notation u r,k = u : The "if" part follows by showing that any sequence [M, f, , it easily follows that, for every q and k, we have sup In fact, arguing inductively on q, it is easy to see that we can assume that each f ′ q+1,p(q+1,ℓ) is a subsequence of f ′ q,p(q,ℓ) , and therefore f ′ q+1 extends f ′ q . Thus the functions f ′ q can be combined to define a function The following is an elementary consequence of Proposition 2.18. Corollary 2.20. Let M be a complete connected Riemannian n-manifold, let H = H 1 ⊕ H 2 be a direct sum decomposition of Hilbert spaces, and let Then Proposition 2.21. Let M be a complete connected Riemannian n-manifold, and let f ∈ C ∞ (M, H). Then the following properties hold: Proof. This holds because the mapping [M ′ , x ′ , f ′ ] → |(∇f ′ )(x ′ )| is well defined and continuous on M n * . Proposition 2.22. In any minimal compact Riemannian foliated space, all leaves without holonomy are repetitive.
Proof. This is a direct consequence of the Reeb's local stability theorem and the fact that L ∩ U is relatively dense in L for all leaf L and open U = ∅ in a minimal compact foliated space [8, Second proof of Theorem 1.13, p. 123].
Example 2.23. For any compact C ∞ foliated space X, there is a C ∞ embedding into some separable Hilbert space, h : X → H [11,Theorem 11.4.4]. Suppose that X is transitive and without holonomy, and endowed with a Riemannian metric. Let M be a dense leaf of X, which is of bounded geometry, and let . Since X is compact and without holonomy, and M is dense in X, it follows from the Reeb's local stability theorem that the leaves of X ′ are the subspaces [L, h| L ], for leaves L of X, and the combination of the corresponding maps mapsι L,h|L is an isometric foliated surjective map ι X,h : X → X ′ . Using that evι X,h = h, we get thatι X,h : X → X ′ is an isometric foliated diffeomorphism, and ev : X ′ → H is a C ∞ embedding whose image is h(X). Thus X ′ is compact and without holonomy, and (M, f ) is limit aperiodic. If moreover X is minimal, then (M, f ) is repetitive by Proposition 2.22.
2.6. The spaces G * and G * . As auxiliary objects, we will use connected (simple) graphs with finite vertex degrees, as well as their (vertex) colorings. For convenience, these graphs are identified with their vertex sets equipped with the natural N-valued metric. This metric is defined as the minimum length of graph-theoretic paths (finite sequences of contiguous vertices) between any pair of points. The existence of geodesic segments (minimizing graph-theoretic paths) between any two vertices is elementary. For such a graph X, the degree of a vertex x is denoted by deg X x (or deg x). The supremum of the vertex degrees is called the degree of X, denoted by deg X ∈ N ∪ {∞}.
Given a countable set F , any map φ : X → F is called an (F -) coloring of X, and (X, φ) is called an (F -) colored graph. We will take F = Z + or F = {1, . . . , c} (c ∈ Z + ). For a connected subgraph Y ⊂ X, we will use the notation ( . These sets form a base of entourages of a uniformity on G * , which is metrizable because this base is countable since U R = U ⌊R⌋ . Moreover it is easy to see that this uniformity is complete. Equip G * with the corresponding underlying topology. The evaluation map ev : , and the degree map deg : are well defined and locally constant. The space G * is also separable; in fact, a countable dense subset of G * is defined by the finite pointed colored graphs because F is countable. Therefore G * is a Polish space. Let (X, φ) be a connected colored graph with finite vertex degrees, whose group of color-preserving graph automorphisms is denoted by Aut(X, φ). There is a canonical mapι X,φ : , can be identified with Aut(X, φ)\X, and has an induced connected colored graph structure. All possible sets [X, φ] form a canonical partition of G * . Like in Lemma 2.14, it follows that the saturation of any open subset of G * is open, and therefore the closure of any saturated On the other hand, (X, φ) (or φ) is called repetitive if, for any p ∈ X and R ≥ 0, the points x ∈ X such that there is a pointed color-preserving graph isomorphism (D X (p, R), p, φ) → (D X (x, R), x, ψ) form a relatively dense subset of X. Clearly, this property is independent of the choice of p. Like in Proposition 2.16, if (X, φ) is repetitive, then [X, φ] is minimal, and the reciprocal also holds when [X, φ] is compact.
There are obvious versions without colorings of the above definitions and properties, which can be also described by taking F = {1}. Namely, we get: a Polish space G * , canonical continuous maps ι X : , whose images, denoted by [X], define a canonical partition of G * , and the concepts of nonperiodic (or aperiodic), limit aperiodic and repetitive graphs. The forgetful (or underlying) map u : We will use the following graph version of (m, R, λ)-p.p.q.i. (Section 2.3). For R ≥ 0 and λ ≥ 1, an (R, λ)pointed partial quasi-isometry (shortly, an (R, λ)-p.p.q.i.) between pointed graphs, (X, x) and (Y, y), is a λ-bilipschitz pointed partial map h : (X, x) (Y, y) such that D(x, R) = dom h, and therefore D(y, R/λ) ⊂ im h. This definition satisfies the obvious analogue of Proposition 2.7. The following is a simple consequence of the fact that graph metrics take integer values.
is repetitive if and only if, given any p ∈ X, for all R > 0 and 1 < λ < 2, the set 9 Each connected graph with finite vertex degrees is countable, and therefore it may be assumed that its underlying set is contained in N. With this assumption, G * is a well defined set. [1,Section A.4]. In other words, this convergence also means that, for all m ∈ N, R > ε > 0 and λ > 1, there is some (m, R, λ)-p.p.q.i. h i : (M, x) (M i , x i ), for i large enough, such that: The  (a) and (b). Like in [1,Theorem A.17], it can be probed that this convergence defines a Polish topology on CM n * , and the forgetful or underlying map u : , whose images, denoted by [M, C, φ], form a canonical partition of CM n * satisfying the obvious versions of Lemmas 2.14 and 2.15. Similarly, the concepts of (limit ) aperiodicity and repetitivity have obvious versions for (M, C, φ), satisfying the obvious version of Proposition 2.16.

Repetitive Riemannian manifolds
Let M be a complete connected Riemannian manifold and fix a distinguished point p ∈ M . For i ∈ N, R > 0, and λ ≥ 1, let Suppose that M is repetitive; i.e., the sets Ω(i, R, λ) are relatively dense in M . We will hereafter consider sequences 0 < r i , s i , t i ↑ ∞ and λ i ↓ 1 satisfying a list of conditions that can be achieved by assuming that these divergences and convergence are fast enough. For integers i, j ≥ 0, we will use the notation For notational convenience, let also r −1 = s −1 = t −1 = ω −1 = 0, and fix any 10 An empty product is assumed to be 1.
Proof. We will prove the first inequality, the proof of the second one being similar.
We have to show that On the other hand, (3.6) yields Now (3.9) follows from (3.10) and (3.11).
, and let ≤ denote its reflexive closure. We will prove that ≤ is in fact a partial order relation (Lemma 3.3 (b)). Let P j i denote the set of maximal elements of (P j i , ≤), which is nonempty because all chains in P j i are finite. Proposition 3.2. For all integers 0 ≤ i < j, there is a set 11 M j i = M j i ∪ · M j i ⊂ M and, for every x ∈ M j i , there is an (i, r i , Λ i,j−1 )-p.p.q.i. h j i,x : (M, p) (M, x) satisfying the following properties: 11 The dotted union symbol denotes a union of disjoint subsets.
Here, the last inclusion is true since, for all y ∈ D(x ′ , Λ i,j−1 r i ), (a) For (l, z), (l, z ′ ) ∈ P j i , any of the following properties yields z = z ′ : is a partially ordered set. Proof. Let us prove (a). It is obvious that (III) yields z = z ′ since ≤ is the reflexive closure of <. Item (I) implies (II) because, since i < l, we get 2r l + 2s i < s l /λ 5 0 < s l /Λ l+1,j−1 by (3.2) and (3.7). According to (4.1), we have l > i and z, z ′ ∈ M j l , so (II) yields z = z ′ because M j l is s l /Λ l+1,j−1 -separated by the induction hypothesis.
there is a unique (l, z) ∈ P j i such that x ∈ h j l,z (D(p, r l )). In particular, for all (k, y) ∈ P j i , there is a unique (l, z) ∈ P j i satisfying (k, y) ≤ (l, z).
But now (3.1) yields Let us define the disjoint sets M j i and M j i , whose union is the definition of M j i . First, let (3.14) Note that this set is well-defined since M l i ⊂ D(p, r l ) = dom h j l,z by the induction hypothesis with (ii). Second, take any maximal s i -separated subset By Lemma 3.4 (b), there is a unique (l, z) ∈ P j i such that x ∈ h j l,z (D(p, r l )). Then define h j i, , as explained in Remark 1.
Lemma 3.5. If (l, z) ∈ P j i , then z ∈ M j l . Proof. The statement is true for l = j − 1 because M j j−1 = M j j−1 by definition. Suppose by absurdity that l < j − 1 and z ∈ M j l . Then, by (3.14), there is some (l ′ , z ′ ) ∈ P j i with l ′ > l and z ⊂ h j l ′ ,z ′ (M l ′ l ). Thus (l, z) < (l ′ , z ′ ), a contradiction.
Lemma 3.6. The following properties hold for every x ∈ M j i : Proof. Item , and (b) follows from Lemma 3.5. Finally, (c) follows from the equality h j i,x = h j l,z h l i,x ′ , the induction hypothesis and Proposition 2.7.
Once we have made the relevant definitions, let us show that they satisfy the properties listed in Proposition 3.2. Item (i) is guaranteed by the definition of M j i , so we really start by proving (ii). The inclusion M j i ⊂ Ω i,j−1 is obvious by Lemma 3.6 (c). Let us prove that M j i ⊂ D(p, r j − t i ). We have M j i ⊂ D(p, r j − t i ) by construction, so let us show that M j i ⊂ D(p, r j − t i ). By the induction hypothesis with (ii), we have z ∈ D(p, r j − t l ) for all (l, z) ∈ P j i . Then D(z, λ l r l ) ⊂ D(p, r j − t i ) because, for any y ∈ D(z, λ l r l ), d(y, p) ≤ d(y, z) + d(z, p) < λ l r l + r j − t l < r j − t i by (3.3). Thus M j i ⊂ D(p, r j − t i ) according to (3.14), since h j l,z : (M, p) (M, z) is an (l, r l , λ l )-p.p.q.i. for all (l, z) ∈ P j i by Lemmas 3.5 and 3.6, and M l i ⊂ D(p, r l ) by the induction hypothesis with (ii). The proof of (ii) is concluded by showing that M j i is s i /Λ i,j−1 -separated. To begin with, we prove that by the induction hypothesis with (ii). By (3.14), it is enough to show that d(z, z ′ ) ≥ Λ l,j−1 r l + Λ l ′ ,j−1 r l ′ + s i /Λ i,j−1 (3.16) for (l, z) = (l ′ , z ′ ) in P j i . If l = l ′ , then, by (3.2) and (3.7), Thus (3.16) follows from the induction hypothesis with (ii) applied to M j l . If l < l ′ , then (3.2) yields s l ≥ λ 0 (r l + s i ) ≥ Λ l,j−1 r l + s i /Λ i,j−1 .
The set M j i is s i -separated by construction. Thus, to prove that (3.14) and (3.15), there is some (l, z) ∈ P j i such thatx ∈ D(z, Λ l,j−1 r l ) andx / ∈ D(z, λ l Λ l,j−1 (r l + s i )). By the triangle inequality, we get d(x,x) ≥ s i , which concludes the proof of (ii).
Let us prove (iii). Let (l, z) ∈ P j i and x ∈ M j i ∩ h j l,z (D(p, r l )). We have by (3.15) and Lemma 3.4 (b),(c), and therefore x ∈ M j i . Consider first the case where (l, z) ∈ P j i . Then the equality h j i,x = h j l,z h l i,x ′ , for x ′ = (h j l,z ) −1 (x), is precisely the definition of h j i,z . Therefore we can suppose that (l, z) ∈ P j i \ P j i . According to Lemma 3.4 (b), there is a unique (l ′ , z ′ ) ∈ P j i such that (l, z) < (l ′ , z ′ ) and x ∈ im(h j l ′ ,z ′ ). We have already proved that h j i, Let us prove (iv). Let (l, z) ∈ P j i . By (3.18), we only have to show that and M l i ⊂ D(p, r l ), yielding (3.19). Suppose now that (l, z) ∈ P j i \ P j i . Then, according to Lemma 3.4 (b), there is a unique (l ′ , z ′ ) ∈ P j i such that (l, z) < (l ′ , z ′ ). We have already proved that Let y = (h j l ′ ,z ′ ) −1 (z). By the induction hypothesis with (iv), we know that M l ′ i ∩ h l ′ l,y (D(p, r l )) = h l ′ l,y (M l i ). Thus (3.19) follows using (iii): , completing the proof of (iv).
Let us prove (v). If (l, z) ∈ P j i , then the result follows from (3.14) and (3.15). So suppose (l, z) / ∈ P j i . Consider first the case where x ∈ M j i . By Lemma 3.4 (b), there is a unique (l ′ , z ′ ) ∈ P j i such that (l, z) < (l ′ , z ′ ), and therefore Lemmas 3.5 and 3.6, and (ii) give
finishing the proof of (v).
. By (ii) and the induction hypothesis with (vii), we have z ∈ B(p, r j−1 ), p ∈ M j j−1 and h j j−1,p = id B(p,rj−1) . By the definitions of P j i in (4.1) and <, it is immediate that (j − 1, p) ∈ P j i . Then Remark 3. Note that, in the course of the proof of Proposition 3.2, the only properties needed from the sets Ω i are the inclusions Ω i ⊂ Ω(i, r i , λ i ) and the fact that Ω i is relatively dense in M . Therefore Proposition 3.2 also holds by substituting the sets Ω i with a prescribed family of subsets of M satisfying the above conditions, after possibly changing the constants ω i . Similarly, the choice of (i, r i , λ i )-p.p.q.i. h j i,x for x ∈ M j i is arbitrary. So, if we have for every x ∈ Ω i a prescribed (i, r i , λ i )-p.p.q.i. f x : (M, p) (M, x), then we can also assume that For every x ∈ M i , there is some j ≥ i such that x ∈ M j i . Then let h i,x = h j i,x , which is independent of j by Proposition 3.2 (vi). Thus the order relations ≤ on the sets P j i (j ≥ i) fit well to define an order relation ≤ on P i ; more precisely, ≤ is the reflexive closure of the relation < on P i defined by setting (j, The following result is a direct consequence of Proposition 3.2. (iv) For every (j, y) ∈ P i and x ∈ M i ∩ h j,y (D(p, r j )), we have h i,x = h j,y h i,x ′ , where x ′ = h −1 j,y (x). (v) For any x ∈ M i and (j, y) ∈ P i , either d(x, z) ≥ λ l (r l + s i ), or x ∈ h l,z (M l i ). (vi) For i ≤ j, we have M j ⊂ M i , and h i,x = h j,x | D(p,ri) for x ∈ M j . (vii) We have p ∈ M i and h i,p = id D(p,ri) .

For integers 0 ≤ i < j, let
the lemma follows by proving that, for all (l, z) ∈ P j i ,  .7), and (3.22) follows.
Proof. Let z ∈ I j i . Since Ω i is ω i -relatively dense in M , there is some y ∈ Ω i with d(y, z) ≤ ω i . Thus by (3.21). Then, by Proposition 3.2 (i), the set M j i ∪{y} cannot be s i -separated and properly contain M j i . So, either y ∈ M j i , or there is some x ∈ M j i \ {y} with d(x, y) < s i . In the former case, d(z, M j i ) ≤ d(z, y) ≤ ω i , whereas, in the latter, For i ∈ N, let Proof. Let x ∈ M . We have D(x, ω i ) ⊂ D(p, r j − t i ) for j large enough. If x / ∈ I i , then x / ∈ h j,p (I j i ) = I j i . So, according to (3.21), there is some (l, z) ∈ P j i such that x ∈ D(z, λ l Λ l,j−1 (r l + s i ) + ω i ) . (3.23) We can suppose that (l, z) minimizes d(x, z) among the elements in P j i satisfying (3.23). Moreover we can assume that l is the least value such that (l, z) is in P j i and satisfies the above properties. Consider first the case where x / ∈ h l,z (B(p, r l − t i − ω i )). Let τ : [0, 1] → M be a minimizing geodesic segment with τ (0) = x and τ (1) = z. There is some a ∈ [0, 1) such that where the last inclusion holds because h l,z : (M, p) (M, z) is an (l, r l , Λ l )-p.p.q.i. Then, by (3.23), (3.5) and (3.7), and then the result follows from Lemma 3. 10.
for some (l ′ , z ′ ) ∈ P l i , according to (3.21). Assume first z ′ = p, and let us prove that z = p. Suppose by absurdity that z = p. We have h −1 l,z (x) = x by Proposition 3.9 (vii). So (3.24) gives , contradicting our choice of (l, z) because l ′ < l.
Since p ∈ M l ′ by Proposition 3.9 (vii), we have d(p, z ′ ) ≥ s l ′ /Λ l ′ +1 by Proposition 3.9 (i). So, by (3.24), z) is an (l, r l , Λ l )-p.p.q.i., and using (3.25), (3.2), (3.5), (3.7) and (3.24), it follows that This contradicts the assumption that (l, z) minimizes d(z, x) because (l ′ , h l,z (z ′ )) ∈ P j i . At this point, only the case z ′ = p remains to be considered. Then, since h l,z : (M, p) (M, z) is an (l, r l , Λ l )-p.p.q.i., and using (3.24), (3.5) and (3.7), we get Note that Proposition 3.2 (vi) yields (l ′ , z) ∈ P j i since i < l ′ < l. Thus the minimality of l gives using (3.23). Then, arguing like in the second paragraph of the proof, we construct a minimizing geodesic segment τ from x to z that meets h l ′ ,z (S(z, r l ′ − t i − ω i )) at a point τ (a) satisfying Using Lemma 3.1, we get and then the result follows from Lemma 3.10.
). By Lemma 3.12, it is enough to show that M i is relatively dense in I i . Let y ∈ I i . By definition of I i , there is some (l, z) ∈ P i such that y ∈ h l,z (I l i ). By Lemma 3.11, there is some x ∈ M l i ⊂ dom h l,z such that d(h −1 l,z (y), x) ≤ ω i + s i . By Proposition 3.9 (iii), we have h l,z (x) ∈ M i . Then the fact that h l,z : (M, p) (M, z) is an (l, r l , Λ l )-p.p.q.i. gives Proposition 3.14. For every η > 0, there is a separated η-relatively dense subset X ⊂ M such that p ∈ X, and, for all (l, z) ∈ P 0 , X ∩ h l,z (D(p, r l )) = h l,z (X ∩ D(p, r l )) . (3.26) We will derive this result from the following auxiliary lemma.
Lemma 3.15. For any η > 0 and 0 < δ < η/Λ 1 , there are sets X 1 ⊂ X 2 ⊂ · · · ⊂ M containing p such that: (a) every X i is δ/Λ 1,i−1 -separated and δΛ 1,i−1 -relatively dense in D(p, r i ); and, Proof. We proceed by induction on i ∈ Z + . Let X 1 be a maximal δ-separated subset of D(p, r 1 ) containing p, given by Zorn's lemma. By Lemma 2.3, it is also δ-relatively dense in D(p, r 1 ). Now, given any i > 1, suppose that we have already defined X k for 1 ≤ k < i satisfying (a) and (b). Let Note that X i−1 ⊂ X i by Proposition 3.2 (vii). The following assertion follows from the induction hypothesis with (a) and Proposition 3.9 (ii).
Proof of Proposition 3.14. For any δ < η/Λ 1 , let X be the union of the sets X i given by Lemma 3.15. By Lemma 3.15 (a) and since r i ↑ ∞, this set is δ/Λ 1 -separated and δΛ 1 -relatively dense in M ; in particular, it is η-relatively dense in M because δΛ 1 < η. Finally, (3.26) follows easily from (3.20), Lemma 3.15 (b) and Proposition 3.9 (vii).
Remark 4. According to the proofs of Proposition 3.14 and Lemma 3.15, we can assume the separating constant of X to be any τ < η/Λ 2 1 . Therefore we can take s 1 as large and Λ 1 as close to 1 as desired, and still assume that X is η-relatively dense and τ -separated. This follows because, according to (3.1)-(3.7), enlarging s i only forces Λ 1 to be smaller. Proposition 3.16. In Proposition 3.14, given any σ > 0, we can assume that there is some 0 < ρ < σ such that, for all l ∈ Z + and x, y ∈ X, In particular, d(x, y) / ∈ (σ − ρ, σ + ρ) for all x, y ∈ X.

Now, by Proposition 2.12, there is an
for all x, y ∈ X i and X i ⊂ X i . Letĥ i : Now (3.33) implies (3.30) for all x, y ∈ X i . Letf i : X ′ i → X i be given byh i (y) = h l,z f l (h −1 l,z (y)), where (l, z) is the only element in P i 0 such that y ∈ h l,z (D(p, r l )). By Proposition 3.9 (ii) and the induction hypothesis with (a), this map satisfies d(y,f i (y)) ≤ 3Λ 1,i−1 ε/2 for all y ∈ X ′ i \ X ′ i . The combination off i andf i into a map f i : X ′ i → X i is the desired bijection, and trivially satisfies both (a) and (c). Item (b) follows from (a) and Proposition 2.12, whereas (d) follows from the definition of X i and (3.34), completing the proof of Claim 4.

Repetitive colored graphs
The results of Section 3 have obvious versions for (colored) connected graphs with finite vertex degrees, using (colored) graph repetitivity with respect to pointed partial quasi-isometries and graph-theoretic geodesic segments (Section 2.6). The proofs are essentially the same, omitting the use of m. By Corollary 2.25, taking Λ 0 < 2, we get Ω i = Ω i = Ω i,j , and these sets are independent of the sequence λ i ↓ 1. However the sequence λ i ↓ 1 is still needed because some steps would not work with λ i = 1, like (3.1), (3.6) and (3.8). Note that the version for (colored) graphs of Proposition 3.14 is trivial. The versions for colored graphs of Propositions 3.2, 3.9 and 3.13, and other observations, are explicitly stated here because they will be used in the proof of Theorem 1.1.
Let (X, φ) be a colored connected graph with finite vertex degrees. Fix any p ∈ X. For R > 0, let Ω(R) be the set of elements x ∈ X such that there exists a pointed color-preserving graph isomorphism (D X (p, R), p, φ) → (D X (x, R), x, φ). Suppose that (X, φ) is repetitive; i.e., the sets Ω(R) are relatively dense in X. Take sequences 0 < r i , s i , t i ↑ ∞ and λ i ↓ 1, and let ω i denote the smallest positive real such that Ω i := Ω(r i ) is ω i -relatively dense in X. Let also r −1 = s −1 = t −1 = ω −1 = 0. With the notation of Section 3, suppose that r i , s i , t i and λ i satisfy Eqs. (3.1) to (3.6), and assume Λ 0 < 2. For i ∈ N, let X i i = {p} and h i i,p = id D(p,ri) . In Proposition 4.1, we will continue defining a subset X j i ⊂ X for every 0 ≤ i < j, and a pointed color-preserving graph isomorphism h j i,z : (D(p, r i ), p, φ) → (D(z, r i ), z, φ) for every z ∈ X j i . Using this notation, let Note that P j k ⊂ P j i if i ≤ k < j. Moreover, let < be the binary relation on P j i defined by declaring (l, z) < (l ′ , z ′ ) if l < l ′ and z ∈ h j l ′ ,z ′ (X l ′ l ), and let ≤ denote its reflexive closure, which is a partial order relation (the analogue of Lemma 3.3 (b)). Let P j i denote the set of maximal elements of (P j i , ≤), which is nonempty since all chains in P j i are finite. For every (k, y) ∈ P j i , there is a unique (l, z) ∈ P j i so that (k, y) ≤ (l, z) (the analogue of Lemma 3.4 (b)).
Proposition 4.1. For all integers 0 ≤ i < j, there is a set X j i = X j i ∪ · X j i ⊂ X and, for every z ∈ X j i , there is a pointed color-preserving graph isomorphism h j i,z : (D(p, r i ), p, φ) → (D(z, r i ), z, φ) satisfying the following properties: For all integers 0 ≤ k ≤ l such that either l < j and k ≥ i, or l = j and k > i, we have X l k ⊂ X j i and h j i,z = h l k,z | D(p,ri) for any z ∈ X l k . (vii) We have p ∈ X j i and h j i,p = id D(p,ri) .
For all x ∈ X i , there is some j ≥ i such that x ∈ X j i . Thus let h i,x = h j i,x , which is independent of j by Proposition 4.1 (vi). Hence the order relations ≤ on the sets P j i (j ≥ i) define an order relation ≤ on P i , which is the reflexive closure of the relation < on P i given by setting (j, x) < (j ′ , x ′ ) if j < j ′ and x ∈ h j ′ ,x ′ (X j ′ j ). Proposition 4.2. The following properties hold: For any x ∈ X i and (j, y) ∈ P i , either d(x, z) ≥ r l + s i , or x ∈ h l,z (X l i ). (vi) For i ≤ j, we have X j ⊂ X i , and h i,x = h j,x | D(p,ri) for x ∈ X j . (vii) We have p ∈ X i and h i,p = id D(p,ri) .
Remark 5. Using the same argument as in Remark 3, we can assume that Ω i (i ∈ N) is any family of relatively dense subsets of Ω(r i ), so that X j i ⊂ Ω i . If, for every x ∈ Ω i , we have a prescribed isometry f i,x : D X (p, r i ) → D X (x, r i ), then we may assume that h j i,x = f i,x for every x ∈ X j i . Finally we have that, for every x ∈ X i , the map h j i,x is a composition of the form f iL,xL · · · f i1,x1 by the analogue of Lemma 3.6. The following result is the analogue for colored graphs of Lemma 3.12 Proposition 4.3. X i is relatively dense in X, where the implied constant only depends on r i , s i , t i and ω i . Remark 6. The versions without colorings of the results of this section hold as well; indeed, they can be considered as the particular case of colorings by one color.

5.
Realization of manifolds as leaves 5.1. Realization in compact foliated spaces without holonomy.
Theorem 5.1. For any (repetitive) connected Riemannian manifold M of bounded geometry, there is a (minimal ) compact Riemannian foliated space X without holonomy with a leaf isometric to M .
To prove this theorem, the construction of X begins with the following result.
Proposition 5.2. Let M be a (repetitive) connected Riemannian manifold of bounded geometry. For every η > 0, there is some separated η-relatively dense subset X ⊂ M , and some coloring φ of X by finitely many colors such that (M, X, φ) is (repetitive and ) limit aperiodic.
Claim 5. The graph X is connected, and X ∩ D M (x, r) ⊂ D X (x, ⌊r/η⌋ + 1) for all x ∈ X and r > 0.
Let x, y ∈ X and k = ⌊d(x, y)/η⌋+1. Since M is connected, there is a finite sequence x = u 0 , u 1 , . . . , u k = y such that d M (u i−1 , u i ) < η (i = 1, . . . , k). Using that X is η-relatively dense in M , we get another finite sequence x = z 0 , z 1 , . . . , z k = y in X so that d M (u i , z i ) < η for all i. Then So, either z i−1 = z i , or there is an edge between z i−1 and z i . Thus, omitting consecutive repetitions, z 0 , z 1 , . . . , z k gives rise to a graph-theoretic path between x and y in X. This shows that X is a connected graph and d X (x, y) ≤ k, as desired.
By Proposition 2.11, there is some c ∈ N such that, for all x ∈ M , the disk D M (x, σ) ∩ X has at most c points, obtaining that deg X ≤ c. Now [5,Theorem 1.4] ensures that there exists a limit aperiodic coloring φ : X → {1, . . . , c}. By the definition of the graph structure of X, we also get for all x ∈ X and r ∈ N.
yielding y = z because X is τ -separated. So y is uniquely associated to y ′ , and therefore the assignment y ′ → y defines a color-preserving map (5.4). Since the choice of δ, λ and R was arbitrary, we infer that X ′ is a τ -separated subset of M ′ . In particular,h is injective by (5.3) and (5.4). By taking δ and λ − 1 small enough, we can also assume that These inequalities, (5.4) and the injectivity ofh show that is a color-preserving graph isomorphism. Like in (5.4), for all y ′ ∈ X ′ ∩ D M ′ (x ′ , R − δ), We use these inequalities to show that Here, the second inclusion is a direct consequence of (5.7). To show the first inclusion, observe that , completing the proof of (5.8). Now, for any is an (m, R, λ)-p.p.q.i. Since X is η-relatively dense, there is some y ∈ M such that d(h(y ′ ), y) ≤ η. We have y ∈ D M (x, (R − 2δ)/λ) by the triangle inequality. Moreover y ∈ imh by (5.8). Soh −1 (y) ∈ X ′ and d(y ′ ,h −1 (y)) < d(y ′ ,ȳ) + δ ≤ λd(h(y ′ ), y) + δ ≤ λη + δ .
Since R is arbitrarily large, and δ and λ − 1 are arbitrarily small, it follows that X ′ is η-relatively dense in M ′ , completing the proof of (a).
Item (b) follows from (a) with the same argument as in Claim 5. Finally, (c) and (d) follow using (5.8) and the color-preserving graph isomorphisms (5.6). This completes the proof of Claim 6.

Consider any class [M
. . , c}), and let h be an isometry of M ′ preserving X ′ and φ ′ . Then h defines a color-preserving graph automorphism (X ′ , φ ′ ) with the above graph structure. By Claim 6 and since (X, φ) is limit aperiodic, we get that h = id on X ′ . By Proposition 2.13, it follows that h = id on M ′ if η is small enough. So (M ′ , X ′ , φ ′ ) is aperiodic, completing the proof of Claim 7. Now assume that M is repetitive, and take the separated η-relatively dense subset X ⊂ M given by Proposition 3.14. Moreover assume that X satisfies the additional conditions of Proposition 3.16 for any given σ ≥ 3η and with some 0 < ρ < σ. Define a graph structure on X using σ and ρ like in the previous case. According to Proposition 3.14, for every (l, z) ∈ P 0 , we have a pointed bijection for every (l, z) ∈ P 0 , which are pointed graph isomorphisms by (3.30) in Proposition 3.16. As before, the graph X is connected, there is some c ∈ N such that deg X ≤ c, there is a repetitive limit aperiodic coloring φ : X → {1, . . . , c}, and (M, X, φ) is limit aperiodic if η is small enough. Let us prove that we can assume that (M, X, φ) is repetitive in this case. To construct φ and prove its limit aperiodicity and repetitivity, the argument of [5,Theorem 1.4] for all x ∈ X and i ∈ N. So, according to [5,Remark 2], the versions without colorings of Propositions 4.1 and 4.2 hold with the maps induced by the pointed graph isomorphisms (5.9). Then the proof of [5,Theorem 1.4] describes the repetitivity of the colored graph (X, φ) using the pointed graph isomorphisms (5.10). By Claim 5, any sequence Proof. Take r 0 > 0 and normal parametrizations κ x : B r0 → B M (x, r 0 ) (x ∈ M ) like in Proposition 2.10. For any 0 < r < r 0 , take X, c and φ like in Proposition 5.2 with η = 2r/3. Write X = { x i | i ∈ I } for some index set I, and let κ i = κ xi : B r → B M (x i , r) and φ i = φ(x i ) (i ∈ I). Consider the graph structure on X defined in the proof of Proposition 5.2, using σ = 3η = 2r. Since deg X ≤ c, there is a coloring α : X → {1, . . . , c + 1} such that adjacent vertices have different colors. Let X k = α −1 (k) and I k = { i ∈ I | x i ∈ X k } (k = 1, . . . , c + 1).
For n = dim M , let S be an isometric copy in R n+1 of the standard n-dimensional sphere so that 0 ∈ S. Choose some function ρ ∈ C ∞ c (R n ) such that ρ(x) depends only on |x|, 0 ≤ ρ ≤ 1, ρ(x) = 1 if |x| ≤ r/2, and ρ(x) = 0 if |x| ≥ r. Take also some C ∞ map τ : R n → R n+1 that restricts to a diffeomorphism B r → S \ {0} and maps R n \ B r to 0.
be the extension by zero of the combination of the compactly supported functions ( Claim 8. If r is small enough, then f is limit aperiodic. By the definition of f k 2 , it follows that there is a sequence i a,q ∈ I k such that d M (x ia,q ,x a,q ) → 0. Given 0 < θ < r/2, we get h q (D M ′ (x ′ a , θ)) ⊂ B M (x ia,q , r/2) for q ≥ p large enough, and κ −1 . Now, using the properties of X and the convergence d M (x ia,q ,x a,q ) → 0, it easily follows that X ′ is also separated and η-relatively dense in M ′ , and, for all x ′ ∈ M ′ , the ball B M ′ (x ′ , σ) ∩ X ′ has at most c points. Hence, like in the case of X, the set X ′ becomes a connected graph with deg X ′ ≤ c by attaching an edge between x ′ a and From the convergence d M (x ia,q ,x a,q ) → 0, we also get that, if p and q are large enough with q ≥ p, then, for all a, b ∈ A with x ′ a , x ′ b ∈ D p , there is an edge in X between x ia,q and x i b,q if and only if there is an edge in X ′ between x ′ a and x ′ b . Thus an injectionh p,q : D p → X is defined byh p,q (x ′ a ) = x ia,q , andh p,q : D p →h p,q ( D p ) is a graph isomorphism. Moreover, for any N ∈ Z + and a ∈ A, we have for all k = 1, . . . , c + 1 and j = 1, 2. So h(X ′ ) = X ′ and h : X ′ → X ′ is a graph isomorphism preserving φ ′ . Since (X ′ , φ ′ ) is aperiodic, it follows that h is the identity on X ′ . So h = id on M ′ if r is small enough by Proposition 2.13. This completes the proof of Claim 8. When M is repetitive, the repetitivity of f is a direct consequence of the repetitivity of (M, X, φ).
Theorem 5.4. For any (minimal ) transitive compact C ∞ foliated space X without holonomy, there is a C ∞ (minimal ) matchbox manifold M without holonomy, and there is a C ∞ surjective foliated map π : M → X that restricts to diffeomorphisms between the leaves of M and X.
Proof. Fix any dense leaf M of X, an auxiliary Riemannian metric on X, and a C ∞ embedding into some separable Hilbert space, h : X → H 1 . Let f 1 = h| M and M 1 = [M, f 1 ] in M n * (H 1 ) (n = dim M ). Then (M, f 1 ) is limit aperiodic, M 1 is compact, and we have an induced isometric diffeomorphism between Riemannian foliated spaces,ι X,h : X → M 1 (Example 2.23).
There are regular foliated atlases and every T i is a relatively compact subspace of T i . Moreover the projectionsp i = pr 2φ i : U i → T i extend the projections p i = pr 2 φ i : U i → T i , and the elementary holonomy transformationsh ij : Let I denote the set of all finite sequences of indices in {1, . . . , c}. For every I = (i 0 , i 1 , . . . , i k ) ∈ I, leth I =h i k−1 i k · · ·h i1i0 and h I = h i k−1 i k · · · h i1i0 , which may be empty maps. There are points y i ∈ B i such that the local transversalsφ −1 i ({y i } × T i ) ≡ T i have disjoint closures in X, and therefore we can realize T := i T i as a complete transversal in X (Section 2.4). Hence φ −1 i ({y i } × T i ) ≡ T i and T := i T i also have these properties.
Since X is Polish and compact, it is locally compact and second countable, and therefore T is also locally compact and second countable. Then there is a countable base of relatively compact open subsets V k (k ∈ N) of T. Fix any relatively compact open subset S i of every T i containing T i , and let S = i S i . Given a metric on T inducing its topology, we can suppose that there is a sequence 0 = k 0 < k 1 < · · · in N such that the sets V km , . . . , V km+1−1 cover S and have diameter < 1/(m + 1) for all m ∈ N. Using K = {0, 1} N as a model of the Cantor space, let ψ : T → K be defined by Since I is countable, K I is homeomorphic to K. Let Ψ : T → K I be the map defined by where 0 ≡ (0, 0, . . . ) ∈ K. Observe that Ψ(x) determines Ψh I (x) for all x ∈ T and I ∈ I with x ∈ domh I .
Claim 9. For any sequence x a in S, if ψ(x a ) is convergent in K, then x a is convergent in T, and lim a x a depends only on lim a ψ(x a ).
The convergence of ψ(x a ) in K means that, for every m ∈ N, there is some a m ∈ N such that ψ(x a )(k) = ψ(x b )(k) for all k < k m+1 and a, b ≥ a m . Since the sets V km , . . . , V km+1−1 cover S, it follows that there is a sequence l m ∈ N such that k m ≤ l m < k m+1 and x a ∈ V lm for all a ≥ a m . Thus the limit set k { x a | a ≥ a m } is a nonempty subset of m V lm , which consists of a unique point of S because every V lm is compact with diameter < 1/(m + 1). Thus x a is convergent in T.
Now let y a be another sequence in S such that ψ(y a ) is convergent in K and lim a ψ(y a ) = lim a ψ(x a ). We have already proved that y a is convergent in T. Moreover, taking a m large enough in the above argument, we also get ψ(y a )(k) = ψ(x a )(k) for all k < k m+1 and a ≥ a m . This yields y a ∈ V lm for all a ≥ a m , and therefore lim a y a = lim a x a . This completes the proof of Claim 9.
According to Claim 9, a continuous map ̟ : For a = 1, 2, let Π a : H → H a denote the corresponding factor projection. Then Π 1 * : M → M 1 is a surjective C ∞ foliated map restricting to isometries between the leaves, and therefore π := (ι X,h1 ) −1 Π 1 * : M → X is also a surjective C ∞ foliated map restricting to isometries between the leaves. Thus every leaf of M is of the form is the first factor projection. Using the description of the C ∞ foliated structure of M n * ,imm (H) given in [4,Section 5], it is easy to check that By the conditions on the functions λ i , this subspace is homeomorphic to the subspace which in turn is homeomorphic to the subspace i Ψ(X i ) ⊂ K I because ̟ is continuous. So T ′ and T ′ are zero-dimensional, obtaining that M is a matchbox manifold. Now suppose that X is minimal. Then (M, f 1 ) is repetitive (Example 2.23). A simple refinement of the proof of Proposition 2.22 also shows that (M, f 2 ) is repetitive. In both cases, this property can be described with the same partial pointed quasi-isometries given by the Reeb's local stability theorem. So (M, f ) is also repetitive, and therefore M is minimal by Proposition 2.16 (i).
As explained in Section 1.4, Theorem 1.1 is a direct consequence of Theorems 5.1 and 5.4.

5.3.
Attaching flat bundles to foliated spaces. Let X ≡ (X, F) be a compact C ∞ foliated space of dimension n, and let M be a leaf of X. On the other hand, let ρ : E → M be a locally compact flat bundle with typical fiber F and horizontal foliated structure H. It can be described as the suspension of its holonomy homomorphism h : π 1 M → Homeo(F ), whose image is its holonomy group G; they are well defined up to conjugation in Homeo(F ). Any foliated concept of E refers to H. The C ∞ differentiable structure of M induces a C ∞ differentiable structure of H. Assume that F is a non-compact locally compact Polish space; then E also has these properties. The notation E x = ρ −1 (x) and E X = ρ −1 (X) will be used for x ∈ M and X ⊂ M .
The one-point compactifications E + x = {x} ⊔ E x of the fibers E x (x ∈ M ) are the fibers of another C ∞ flat bundle ρ + : E + → M ; thus E + ≡ M ⊔ E as sets. Its typical fiber is the one-point compactification F + = {∞} ∪ F of F , the leaves of its horizontal foliation H + are M and the leaves of H, its holonomy homomorphism h + : π 1 M → Homeo(F + ) is induced by h, and its holonomy group is denoted by G + . The more specific notation h x : π 1 (M, x) → Homeo(F ), h + x : π 1 (M, x) → Homeo(F + ), G x and G + x will be used to indicate the base point x.
Let X ′ = X ⊔ E, equipped with the following topology. Take any foliated chart U ≡ B × T of X, for some ball B ⊂ R n and some local transversal T. We have M ∩ U ≡ B × D for some countable subset D ⊂ T. Since the plaques of U are contractible, ρ has a local trivialization E M∩U ≡ (M ∩ U ) × F of flat bundle. Let T ′ = T ⊔ (D × F ), endowed with the topology with basic open sets of the form where z runs in D, R z and S z are open in F , R z is compact for all z, R z = ∅ for all but finitely many z, F \ S z is compact for all z, and S z = F for all but finitely many z. Then X has a topology with basic open sets of the form for all possible foliated charts U ≡ B × T of X. Using these basic open sets, it is easy to check that X ′ is Hausdorff, second countable and compact. So X ′ is metrizable [32,Proposition 4.6], and hence Polish. In particular, the sets U ′ = U ⊔ E M∩U ≡ (B × T) ⊔ (B × D × F ) = B × T ′ are open in X ′ , and the fibers B × { * } correspond to open subsets of leaves of F or H. Thus these identities are foliated charts of a foliated structure F ′ on X ′ , and its leaves are the leaves of F and H. As sets, we can write X ′ ≡ X ∪ id M E + and T ′ ≡ T ∪ id D (D × F + ), where we consider D ≡ D × {∞} ⊂ D × F + ; we can also write T ′ = T ⊔ E D ≡ T ∪ idD E + D . Consider a regular foliated atlas of X consisting of charts U i ≡ B i × T i , for balls B i ⊂ R n and local transversal T i . As before, take local trivializations E M∩Ui ≡ (M ∩ U i ) × F of the flat bundle ρ, write M ∩ U i ≡ B i × D i for countable subsets D i ⊂ T i , and consider the induced foliated charts , endowed with Polish topologies. The changes of coordinates of the foliated charts U i ≡ B i × T i are of the form (y, z) → (f ij (y, z), h ij (z)), where every mapping y → f ij (y, z) is C ∞ with all of its partial derivatives of arbitrary order depending continuously on z. Using local trivializations of E and foliated charts of F, we get E M∩Ui ≡ (M ∩ U i ) × F ≡ B i × D i × F . The changes of these local descriptions are of the form (y, z, u) → (f ij (y, z), h ij (z), g ij (z, u)), where the maps g ij are independent of y by the compatibility with H. Then the changes of coordinates of the foliated charts Thus the charts U ′ i ≡ B i × T ′ i define a C ∞ structure on X ′ ≡ (X ′ , F ′ ). The corresponding elementary holonomy transformations h ′ ij are combinations of maps h ij and g ij . Using these foliated charts, it also follows that X and E are embedded C ∞ foliated subspaces of X ′ , E + is an injectively immersed C ∞ foliated subspace of X ′ , and the combination π : X ′ → X of id X and ρ (or ρ + ) is a C ∞ foliated retraction. The fibers of π are Lemma 5.5. Suppose that the restrictions of ρ to the leaves of H are regular coverings of the leaves of F, and that the leaf M of F has no holonomy. Then the holonomy group of the leaf M of F ′ is isomorphic to the group of germs at ∞ of the elements of the subgroup G + ⊂ Homeo(F + ).
Proof. With the above notation, fix an index i 0 and some point x 0 ∈ D i0 ≡ T i0 ∩ M ≡ T ′ i0 ∩ M , considering T i0 ⊂ X and T ′ i0 ⊂ X ′ . Let c : [0, 1] → M be a loop based at x 0 . Since the holonomy group of M in X is trivial, there is a family of leafwise loops c x : [0, 1] → X, depending continuously on x in some open neighborhood T 0 of x 0 in T i0 , such that c x0 = c. Let D 0 = D i0 ∩ T 0 . From the above description of the elementary holonomy transformations h ′ ij , it follows that the holonomy of F ′ defined by [c] ∈ π 1 (M, x 0 ) is the germ at x 0 ≡ (x 0 , ∞) of the homeomorphism g c of T ′ i0 = T i0 ⊔ (D i0 × F ) given by using [c x ] ∈ π 1 (M, x). Since the restrictions of ρ to the leaves of H are regular coverings of M , we easily get that h + x ([c x ])(u) = u for some x ∈ D 0 and u ∈ F + close enough to ∞ if and only if h + x0 ([c])(u) = u for u ∈ F + close enough to ∞. So, by restricting every g c to {x 0 } × F + ≡ F + , we get an isomorphism from the holonomy group of the leaf M of F ′ at x 0 to the group of germs of the elements of G + x0 at ∞. Proofs of Corollaries 1.2 and 1.3. Let M be non-compact connected Riemannian manifold of bounded geometry. By Theorem 1.1, M is isometric to a leaf in some Riemannian matchbox manifold M without holonomy. Now Corollaries 1.2 and 1.3 follow by considering the foliated space M ′ constructed as above with M and an appropriate flat bundle E over M , and lifting the Riemannian metric of M to M ′ .
In the case of Corollary 1.2, we can use the trivial flat bundle E = M × K over M , where K is the Cantor space. By the density of M in M, it follows that M ′ has a compact zero-dimensional complete transversal T ′ without isolated points, and therefore T ′ is homeomorphic to the Cantor space.
In the case of Corollary 1.3, let Γ denote the group of deck transformations of the given regular covering M of M , equipped with the discrete topology. If Γ is infinite, we can take E = M , whose typical fiber is F = Γ. If Γ is finite, we can take E = M × Z, whose typical fiber is F = Γ × Z. In any case, F is non-compact, and the action of Γ on itself by left translations induces a canonical action of Γ on F , which in turn induces an action on F + . By Lemma 5.5 and the regularity of the covering M of M , the holonomy group of M in M ′ is isomorphic to the group of germs at ∞ of the action of the elements of Γ on F + , which is itself isomorphic to Γ.