#### Volume 21, issue 2 (2021)

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Connectedness properties and splittings of groups with isolated flats

### G Christopher Hruska and Kim Ruane

Algebraic & Geometric Topology 21 (2021) 755–800
##### 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.

##### Keywords
nonpositive curvature, isolated flats, locally connected, tree of metric compacta
##### Mathematical Subject Classification 2010
Primary: 20E08, 20F67
##### Publication
Received: 4 August 2019
Revised: 17 December 2019
Accepted: 18 May 2020
Published: 25 April 2021
##### Authors
 G Christopher Hruska Department of Mathematical Sciences University of Wisconsin-Milwaukee Milwaukee, WI United States Kim Ruane Department of Mathematics Tufts University Medford, MA United States