Finite cuts and CAT(0) boundaries

TL;DR

This paper investigates the structure of groups acting on CAT(0) spaces, revealing conditions under which the group is either virtually a surface group or splits over a 2-ended group, based on boundary separation properties.

Contribution

It establishes a connection between boundary separation points in CAT(0) spaces and algebraic properties of the acting group, including splitting and virtual surface group classification.

Findings

Groups with boundary separated by m points are either virtually surface groups or split over 2-ended groups.

Nesting actions on -trees with non-overlapping translation intervals induce isometric actions.

The study links boundary topology with algebraic group decompositions.

Abstract

We show that if a 1-ended group $G$ acts geometrically on a CAT(0) space $X$ and $\bd X$ is separated by $m$ points then either $G$ is virtually a surface group or $G$ splits over a 2-ended group. In the course of the proof we study nesting actions on $\R $-trees and we show that nesting actions with non overlapping translation intervals give rise to isometric actions.