Rank-one isometries of proper CAT(0)-spaces

TL;DR

This paper surveys the properties of non-elementary groups acting on proper CAT(0)-spaces, focusing on the dynamics of rank-one isometries and their limit sets, revealing dense orbits and fixed point pairs.

Contribution

It provides a comprehensive analysis of the action of groups containing rank-one elements on the limit set, highlighting new density results and dynamical properties.

Findings

Dense orbit for G on LxL minus the diagonal

Pairs of fixed points of rank-one elements are dense in LxL minus the diagonal

Properties of G's action on the limit set under rank-one assumptions

Abstract

Let G be a non-elementary group of isometries of a proper CAT(0)-space with limit set L. We survey properties of the action of G on L under the assumption that G contains a rank-one element. Among others, we show that there is a dense orbit for the action of G on the complement of the diagonal in LxL and that pairs of fixed points of rank-one elements are dense in the complement of the diagonal of LxL.