A `transversal' for minimal invariant sets in the boundary of a CAT(0) group

TL;DR

This paper introduces new techniques to analyze boundary dynamics of CAT(0) groups, establishing conditions for virtual-abelianity and providing insights into rank rigidity conjecture through boundary sphere intersections.

Contribution

It presents novel methods for studying boundary dynamics of CAT(0) groups, linking boundary sphere intersections to group properties and conjectures.

Findings

Existence of a maximal flat whose boundary intersects all minimal invariant sets

A necessary and sufficient condition for a group to be virtually-Abelian

A new approach to Ballmann's rank rigidity conjecture

Abstract

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal $G$-invariant subset of $\partial_\infty X$. As a result we derive a necessary and sufficient dynamical condition for $G$ to be virtually-Abelian, as well as a new approach to Ballmann's rank rigidity conjecture.