On cyclic CAT(0) domains of discontinuity

TL;DR

This paper studies the behavior of cyclic groups acting by isometries on CAT(0) spaces, focusing on their domains of discontinuity on the boundary, which is crucial for understanding the geometric and dynamical properties of such actions.

Contribution

It provides new insights into the structure of domains of discontinuity for cyclic groups acting on CAT(0) spaces, extending previous understanding of group actions on non-positively curved spaces.

Findings

Characterization of the domain of discontinuity for cyclic groups

Conditions under which the action is properly discontinuous

Implications for the boundary dynamics of CAT(0) spaces

Abstract

Let $X$ be a CAT(0) space, and $G$ a discrete cyclic group of isometries of $X$. We investigate the domain of discontinuity for the action of $G$ on the boundary $\partial X$.