Connectedness properties and splittings of groups with isolated flats

Abstract

We study $CAT\left(0\right)$ groups and their splittings as graphs of groups. For one-ended $CAT\left(0\right)$ groups with isolated flats we prove a theorem characterizing exactly when the visual boundary is locally connected. This characterization depends on whether the group has a certain type of splitting over a virtually abelian subgroup. In the locally connected case, we describe the boundary as a tree of metric spaces in the sense of Świątkowski.

A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary $CAT\left(0\right)$ groups. If a $CAT\left(0\right)$ group splits as a graph of groups with convex edge groups, then the vertex groups are also $CAT\left(0\right)$ groups.