Abstract group actions of locally compact groups on CAT(0) spaces

TL;DR

This paper investigates how locally compact groups act on CAT(0) spaces, establishing conditions for continuity or fixed points, and providing a geometric proof for the continuity of certain group homomorphisms.

Contribution

It extends classical results on group actions to CAT(0) spaces, showing under mild assumptions actions are continuous or have fixed points, and proves a new continuity result for group homomorphisms.

Findings

Actions are continuous under mild conditions

Existence of global fixed points for certain actions

Any homomorphism into a torsion-free CAT(0) group is continuous

Abstract

We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.