# Connectedness properties and splittings of groups with isolated flats

## TL;DR

This paper investigates the boundary properties of CAT(0) groups with isolated flats, providing a characterization of local connectivity based on group splittings, and introduces a convex splitting theorem for CAT(0) groups.

## Contribution

It offers a new criterion for local connectivity of boundaries in CAT(0) groups with isolated flats and establishes a convex splitting theorem for CAT(0) groups with convex edge groups.

## Key findings

- Characterization of local connectivity of boundaries based on group splittings.
- Boundary described as a tree of metric spaces in the locally connected case.
- Convex splitting theorem ensuring vertex groups are CAT(0) when edge groups are convex.

## Abstract

In this paper we study CAT(0) groups and their splittings as graphs of groups. For one-ended CAT(0) 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 \'Swi\k{a}tkowski. A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary CAT(0) groups. If a CAT(0) group splits as a graph of groups with convex edge groups, then the vertex groups are also CAT(0) groups.