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Ping-pong in Hadamard manifolds

Subhadip Dey, Michael Kapovich, Beibei Liu

Abstract

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when $f$ is hyperbolic.

Ping-pong in Hadamard manifolds

Abstract

In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds . Precisely, we prove that a nonelementary discrete isometry subgroup of generated by two non-elliptic isometries , contains a free subgroup of rank generated by isometries of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when is hyperbolic.

Paper Structure

This paper contains 8 sections, 9 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Preliminary material
  4. Some $\mathrm{CAT}(-1)$ computations
  5. Quasiconvex and starlike subsets
  6. Classification of isometries
  7. Margulis cusps and tubes
  8. Displacement estimates

Key Result

Theorem 1.1

There exists a function $\mathcal{L}=\mathcal{L}(n,\kappa)$ such that the following holds: Let $f, g$ be non-elliptic isometries of $X$ generating a nonelementary discrete subgroup $\Gamma$ of $\mathop{\mathrm{\mathrm Isom}}\nolimits(X)$. Then there exists an element $h\in \Gamma$ whose word length

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 3.1
  • Corollary 3.2
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Corollary 3.7
  • ...and 1 more