Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations

Lessa, Pablo

doi:10.4310/ajm.2015.v19.n3.a3

Cited by 15 publications

(20 citation statements)

References 17 publications

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“…The sequence (Π n ) n∈N is called a sequence of convergence mappings of (L n , g n , x n ) n∈N with respect to (L, g, x). This mode of convergence is sometimes called smooth convergence: [29,33]. It appeared first in [25], where Gromov proved that Cheeger's finiteness theorem (see [15]) was in fact a compactness result.…”

Section: Proposition 23 the Lamination L Defined By A Tower T Of Fini...mentioning

confidence: 99%

Topology of leaves for minimal laminations by hyperbolic surfaces

Alvarez¹,

Brum²,

Martínez³

et al. 2019

Preprint

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We construct minimal laminations by hyperbolic surfaces whose generic leaf is a disk and contain any prescribed family of surfaces and with a precise control of the topologies of the surfaces that appear. The laminations are constructed via towers of finite coverings of surfaces for which we need to develop a relative version of residual finiteness which may be of independent interest. The main step in establishing this relative version of residual finiteness is to obtain finite covers with control on the second systole of the surface, which is done in the appendix. In a companion paper, the case of other generic leaves is treated.

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Section: Proposition 23 the Lamination L Defined By A Tower T Of Fini...mentioning

confidence: 99%

Topology of leaves for minimal laminations by hyperbolic surfaces

Alvarez¹,

Brum²,

Martínez³

et al. 2019

Preprint

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“…For any n ∈ N, let M * (n) denote the set 1 of isometry classes, [M, x], of pointed complete connected Riemannian n-manifolds, (M, x). A sequence [17, Appendix A] (see also [81,Chapter 10], [73]). The corresponding Polish space is denoted by M ∞ * (n), and its closure operator by Cl ∞ .…”

Section: Bounded Geometry and Leavesmentioning

confidence: 99%

Generic Coarse Geometry of Leaves

López¹,

Candel²

2018

Lecture Notes in Mathematics

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Contents Chapter 1. Introduction Chapter 2. Coarse quasi-isometries 2.1. Notation, conventions and terminology 2.2. Coarse quasi-isometries 2.3. Coarse composites 2.4. A coarsely quasi-isometric version of Arzela-Ascoli theorem 2.5. Large scale Lipschitz maps 2.6. Coarse and rough maps Chapter 3. Some classes of metric spaces 3.1. Graphs 3.2. Metric spaces of coarse bounded geometry 3.3. Coarsely quasi-symmetric metric spaces 3.4. Coarsely quasi-convex metric spaces Chapter 4. Growth of metric spaces 4.1. Growth of non-decreasing functions 4.2. Growth of metric spaces 4.3. Growth symmetry Chapter 5. Amenability of metric spaces 5.1. Amenability 5.2. Amenable symmetry Chapter 6. Coarse ends 6.1. Ends 6.2. Coarse connectivity 6.3. Coarse ends 6.4. Functoriality of the space of coarse ends 6.5. Coarse end space of a class of metric spaces Chapter 7. Higson corona and asymptotic dimension 7.1. Compactifications 7.2. Higson compactification 7.3. Asymptotic dimension Chapter 8. Pseudogroups 8.1. Pseudogroups 8.2. Coarse quasi-isometry type of orbits 8.3. A version of local Reeb stability iii iv CONTENTS 8.4. Topological dynamics Chapter 9. Generic coarse geometry of orbits 9.1. Coarsely quasi-isometric orbits 9.2. Growth of the orbits 9.3. Amenable orbits 9.4. Asymptotic dimension of the orbits 9.5. Highson corona of the orbits 9.6. Measure theoretic versions Chapter 10. Generic coarse geometry of leaves 10.1. Foliated spaces 10.2. Saturated sets 10.3. Coarse quasi-isometry type of the leaves 10.4. Higson corona of the leaves 10.5. Algebraic asymptotic invariants 10.6. Versions with quasi-invariant currents 10.7. There is no measure theoretic version of recurrence Chapter 11. Examples and open problems 11.1. Foliated spaces defined by suspensions 11.2. Foliated spaces defined by locally free actions of Lie groups 11.3. Inverse limits of covering spaces 11.4. Bounded geometry and leaves 11.5. Graph spaces 11.6. Case of Cayley graphs 11.7. Construction of limit-aperiodic functions 11.8. Graph matchbox manifolds 11.9. Concrete examples of graph matchbox manifolds 11.10. Foliations of codimension one 11.11. Open problems Bibliography CONTENTS vAbstract. A compact Polish foliated space is considered. Part of this work studies coarsely quasi-isometric invariants of leaves in some residual saturated subset when the foliated space is transitive. In fact, we also use "equi-" versions of this kind of invariants, which means that the definition is satisfied with the same constants by some given set of leaves. For instance, the following properties are proved. Either all dense leaves without holonomy are equi-coarsely quasi-isometric to each other, or else there exist residually many dense leaves without holonomy such that each of them is coarsely quasi-isometric to meagerly many leaves. Assuming that the foliated space is minimal, the first of the above alternatives holds if and if the leaves without holonomy satisfy a condition called coarse quasi-symmetry.A similar dichotomy holds for the growth type of the leaves, as well as a...

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“…Example The following simple examples clarify Definition : (i)The Reeb foliation on S 3 with the standard metric is covering‐continuous, but it is not holonomy‐continuous with any Riemannian metric. If the metric is modified around the compact leaf

T^{2} = S^{1} \times S^{1}

so that the diffeomorphism

(x, y) \mapsto (y, x)

of T 2 is not an isometry, then this foliation becomes non‐covering‐continuous. (ii)The Riemannian foliated space of [, Example 2.5] is covering‐determined but not holonomy‐continuous. This example can be easily realized as a saturated subspace of a Riemannian foliated space where the holonomy coverings of the leaves are isometric to