Ping-pong in Hadamard manifolds

TL;DR

This paper establishes a quantitative version of the Tits alternative for negatively pinched Hadamard manifolds, showing conditions under which certain isometry groups contain free subgroups with bounded generators.

Contribution

It proves a quantitative Tits alternative for negatively pinched Hadamard manifolds, identifying conditions for free subgroups with bounded word length generators.

Findings

Existence of free subgroups generated by bounded words

Free subgroups are convex-cocompact when generated by hyperbolic isometries

Quantitative bounds on generators in isometry groups

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.