A strong Schottky lemma on $n$ generators for $\mathrm{CAT}(0)$ spaces

TL;DR

This paper establishes a criterion for $n$ hyperbolic isometries of a $ ext{CAT}(0)$ space to generate a free group, extending previous results for 2 generators and ensuring discreteness under local compactness.

Contribution

It generalizes a known free group generation criterion from 2 to $n$ generators in $ ext{CAT}(0)$ spaces, including conditions for discreteness.

Findings

Provides a new criterion for free group generation in $ ext{CAT}(0)$ spaces.

Extends previous 2-generator results to $n$ generators.

Ensures discreteness of the generated group when the space is locally compact.

Abstract

We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the additional assumption that $X$ is complete and has no fake zero angles. Moreover, when $X$ is locally compact, the group we obtain is also discrete.