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Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

Nicola Cavallucci, Andrea Sambusetti

Abstract

We prove a quantitative version of the classical Tits' alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

Discrete groups of packed, non-positively curved, Gromov hyperbolic metric spaces

Abstract

We prove a quantitative version of the classical Tits' alternative for discrete groups acting on packed Gromov-hyperbolic spaces supporting a convex geodesic bicombing. Some geometric consequences, as uniform estimates on systole, diastole, algebraic entropy and critical exponent of the groups, will be presented. Finally we will study the behaviour of these group actions under limits, providing new examples of compact classes of metric spaces.

Paper Structure

This paper contains 27 sections, 60 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Packed, GCB-spaces
  4. Convex geodesic bicombings
  5. The packing condition and some examples
  6. Margulis constant
  7. Basics of Gromov hyperbolic spaces
  8. Gromov boundary
  9. Projections
  10. Helly's Theorem
  11. Elementary subgroups of packed, $\delta$-hyperbolic spaces
  12. Isometries of packed, $\delta$-hyperbolic GCB-spaces
  13. Special properties in the convex case
  14. The Margulis domain
  15. Free subgroups and sub-semigroups
  16. ...and 12 more sections

Key Result

Theorem 1.1

${}$ Let $P_0,r_0$ and $\delta$ be fixed positive constants. Then, there exists an integer $N(P_0,r_0,\delta)$, only depending on $P_0,r_0, \delta$, satisfying the following properties. Let $(X,\sigma)$ be any $\delta$-hyperbolic, GCB-space which is $P_0$-packed at scale $r_0$: Here by $\sigma$-isometries we mean the natural isometries of $(X, \sigma)$, that is those preserving the geodesic bico

Theorems & Definitions (109)

  • Theorem 1.1: Quantitative free subgroup theorem with specification
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5: Extract from Theorems \ref{['Dicotomia']}& \ref{['elementarita']} and Corollary \ref{['Dicotomia-geometrica']}
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 99 more