A CAT(0) alternative for amenable groups and a Kazhdan-type rigidity principle

TL;DR

This paper establishes a fixed point alternative for finitely generated amenable groups acting on CAT(0) spaces, revealing rigidity phenomena and advancing understanding of torsion groups' actions.

Contribution

It introduces a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property acting on CAT(0) spaces, providing new fixed point and non-action results.

Findings

Amenable groups either have fixed points or admit fixed-point-free actions on certain CAT(0) spaces.

Finitely generated torsion and virtually simple amenable groups cannot act nontrivially on these spaces.

The results progress the understanding of torsion groups' actions on CAT(0) spaces.

Abstract

We prove that finitely generated amenable groups acting on CAT(0) spaces satisfy the following alternative: either every action on a geodesically complete CAT(0) space with bounded geometry (or finite dimension) has a global fixed point, or the group admits a fixed-point-free action on $\mathbb{R}^n$. As a consequence, finitely generated amenable torsion groups and finitely generated virtually simple amenable groups cannot act nontrivially on geodesically complete CAT(0) spaces with bounded geometry or on finite-dimensional complete CAT(0) spaces. The proof relies on a Kazhdan-type rigidity theorem for groups with the Euclidean fixed point property: if such a group acts on a geodesically complete CAT(0) space of bounded geometry with almost fixed points, then it has a genuine fixed point. This yields several further corollaries, including a rigidity dichotomy for drift and that any finitely generated torsion group acting on a geodesically complete visibility CAT(0) space with bounded geometry must have a global fixed point. These results make substantial progress on the longstanding problem of understanding actions of torsion groups on CAT(0) spaces.