Thin actions on CAT(0) spaces

TL;DR

This paper investigates the properties of isometry groups acting on certain CAT(0) spaces with packing constraints, establishing rigidity results and extending the Margulis Lemma to these spaces.

Contribution

It introduces a new rigidity theorem for CAT(0) spaces with packing conditions, generalizing the Margulis Lemma to broader contexts.

Findings

Spaces with negative curvature behavior cannot support thin actions.

The results extend classical rigidity theorems to packed, geodesically complete CAT(0) spaces.

A universal constant depending on packing controls systole bounds.

Abstract

We study groups of isometries of packed, geodesically complete, CAT$(0)$-spaces for which the systole at every point is smaller than a universal constant depending only on the packing, deducing strong rigidity results. We show that if a space as above has some negative curvature behaviour then it cannot support a thin action: this generalizes the classical Margulis Lemma to a broader class of spaces.