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Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces

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Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces G. Besson, G. Courtois, S. Gallot, A. Sambusetti December 2, 2020 Abstract We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy of which we recall the definition. Contents 1 Introduction 2 2 Basic definitions and notations8 3 Entropy, Doubling and Packing Properties 10 3.1 Entropies . 10 3.2 Doubling and Packing Properties . 13 3.3 Comparison between the various possible hypotheses : bound on the entropy, doubling and packing conditions, Ricci curvature . 14 3.4 Doubling property induced on subgroups . 20 4 Free Subgroups 21 4.1 Ping-pong Lemma . 21 arXiv:1712.08386v3 [math.DG] 1 Dec 2020 4.2 When the asymptotic displacement is bounded below . 24 4.3 When some Margulis constant is bounded below: . 27 4.3.1 When the Margulis constant L∗ is bounded below: . 28 4.3.2 When the Margulis constant L is bounded below: . 32 4.4 Free semi-groups for convex distances . 35 5 Co-compact actions on Gromov-hyperbolic spaces 36 5.1 A Bishop-Gromov inequality for Gromov-hyperbolic spaces . 36 5.2 A Tits alternative and lower bounds for the entropies . 41 5.2.1 The alternative: entropy vs asymptotic displacement . 41 5.2.2 Explicit universal lower bounds for the exponential growth . 48 5.2.3 Implicit universal lower bounds for the exponential growth . 51 1 5.3 Margulis Lemmas for group actions on Gromov-hyperbolic spaces . 53 5.3.1 A first Margulis Lemma for δ-hyperbolic spaces . 53 5.3.2 A lower bound of the diastole . 55 5.3.3 A lower bound of the global systole for Busemann spaces . 56 6 Transplantation of Margulis' Properties 60 6.1 The classes of groups which are considered here . 60 6.1.1 Definitions and first properties . 60 6.1.2 Comparison with acylindrical hyperbolicity . 63 6.2 A first Margulis Property . 65 6.3 A lower bound of the diastole . 67 6.4 A lower bound of the global systole . 69 6.5 Structure of thin subsets in quotients of metric measured spaces . 72 6.5.1 Generic topological Lemma . 72 6.5.2 On the topology of thin subsets . 75 6.5.3 Examples and counter-examples . 78 6.5.4 The topology of Margulis' tubes is almost trivial . 82 7 Applications 88 7.1 Application to Manifolds . 88 7.2 Application to Polyhedrons . 91 7.2.1 Some examples . 95 7.3 Finiteness and Compactness Results . 101 7.3.1 General Definitions and Results . 101 7.3.2 Finiteness and compactness results when the Ricci curvature is bounded from below . 102 7.3.3 Finiteness and compactness Results for Einstein Structures . 106 8 Appendix : Basic results on Gromov-hyperbolic spaces 113 8.1 Why is there quantitative gaps between basic sources, and how to precise coher- ently the constants: . 113 8.2 Projections and quasi equality in the triangle inequality . 115 8.3 Asymptotic and diverging geodesic lines . 116 8.4 Isometries and displacements . 118 8.4.1 Discrete subgroups of the group of isometries . 118 8.4.2 Isometries and displacements in Gromov-hyperbolic spaces . 120 8.5 Margulis' domains of an hyperbolic isometry . 126 8.6 Hyperbolic isometries of Busemann Gromov-hyperbolic spaces . 130 8.7 Elementary subgroups . 131 1 Introduction The celebrated Margulis Lemma is the keystone of a beautiful theory of the structure of complete Riemannian manifolds with bounded sectional curvature. It has two main aspects: the first one is algebraic and concerns the fundamental group of the manifold, the second one is more geometric 2 and yields a thin-thick decomposition of the manifold. To be more precise let us state a weak version of this lemma pertaining to the first aspect (see [Mar75], [BGS85], [BZ88] Section 37.3). Theorem 1.1. There exist constants "(n) > 0 and C(n) > 0 such that, for every complete Riemannian manifold M whose sectional curvature satisfies −1 ≤ Sect(M) ≤ 0, every point p 2 M and every " ≤ "(n), the subgroup Γ"(p) of π1(M) generated by the loops at p of length less than " is virtually nilpotent. Furthermore, the index of the nilpotent subgroup is bounded above by C(n). This statement is a weak version, indeed in the strong one the upper bound on the sectional cur- vature could be positive, with extra assumption though. A version of this theorem for manifolds of strictly negative curvature was simultaneously proved by E. Heintze in his habilitationsschrift of 1976 (see [Hei02]). The history of this result goes back to Bieberbach Theorem ([Bie11]) which describes the discrete subgroups of the isometry group of the Euclidean spaces and consequently gives a structure theorem for the flat manifolds and orbifolds. Later, this result was extended to the study of discrete subgroups of Lie groups by H. Zassenhaus ([Zas38]) and to locally symmetric spaces by D. Kazhdan and G. Margulis ([KM69], using Zassenhaus' Lemma). Recently there has been progresses on the question of extending this result to different spaces or curvature conditions: for example, after a short sketch of proof by J. Cheeger and T. Colding (see [CC96]) under the hypothesis \Ricci curvature bounded from below" and a first complete proof of V. Kapovitch, A. Petrunin and W. Tuschmann ([KPT10]) under the hypothesis \sectional curvature bounded from below", V. Kapovitch and B. Wilking ([KW11]) recently established a Margulis-like Lemma under the hypothesis \Ricci curvature bounded from below by −(n−1)" instead of \−1 ≤ Sect(M) ≤ 0" (see also [Cou15] for references and a detailed exposition). This paper is the first of a series of articles devoted to this theme. Here we are interested by both aspects, algebraic and geometric, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions, as mentioned in the title, in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is δ-hyperbolic (in the sense of Gromov, see [Gro87] and Section 8.1 of Appendix8 for precise definitions) and the lower bound by an upper bound on the entropy which we define below. Notice that δ behaves like a distance and it is rather δ−2 which is curvature-like. The starting point of the ideas developed in this paper is the prepublication [BCG03], which initially concerned the isometric actions of a more limited class of groups on less general types of spaces, namely fundamental groups of manifolds with sectional curvature σ ≤ −1 and with injectivity radius ≥ i0 > 0 (and groups such that any non abelian subgroup with two generators admits an injective homomorphism into such a fundamental group). Several developments of the ideas contained in [BCG03] were established by G. Reviron ([Rev08]), F. Zuddas ([Zud11], [Zud09]), F. Cerocchi ([Cer14]), F. Cerocchi and A. Sambusetti ([CS19], [CS17] and [CS16], this last paper being devoted to prove Margulis' properties in the abelian setting) . Let (X; d) be a (non-elementary) metric space which we assume to be proper, i.e the closed metric balls are compact. We only consider metric spaces which are geodesic. More precisely, a geodesic segment is the image of an interval of I ⊂ R by an isometric map from I into X. The space (X; d) is said to be geodesic if any two points of X are joined by at least one geodesic segment. Let µ be a positive (non identically zero) Borel measure. We call (X; d; µ) a measured metric space. Definition 1.2. Let (X; d; µ) be a metric measured space we define its entropy by 1 Ent(X; d; µ) = lim inf ln µ(BX (x; R)) R!+1 R where BX (x; R) is the open ball of radius R around x 2 X. Furthermore, the entropy is inde- pendent of x. 3 In the sequel we will consider a group Γ acting by isometries on (X; d) properly and, often, co-compactly. We recall that the action is said to be proper if for x 2 X and for all R > 0, the number of γ 2 Γ satisfying d(x; γx) ≤ R is finite; this does not depend on x. In that case, for any measure µ invariant by Γ the above definition yields the same number which we call the entropy of (X; d) and denote by Ent(X; d). If (X; d) is a δ-hyperbolic geodesic metric space and γ a torsion-free isometry, we define the asymptotic displacement of γ (sometimes also called \stable displacement") by 1 `(γ) = lim d(x; γkx); k!+1 k this definition does not depend on the choice of x 2 X. One of our results is the following (see Proposition 5.10). Proposition 1.3. For every non-elementary δ-hyperbolic metric space (X; d) and every group Γ acting properly by isometries on (X; d), if diam(ΓnX) ≤ D < +1, then ln 2 ln 2 Ent(X; d) ≥ ≥ ; L + 17δ + 2D 27δ + 10D where L = inf `(γ): γ hyperbolic element of Γ n feg .
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