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(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 Świątkowski.

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.

Keywords
nonpositive curvature, isolated flats, locally connected, tree of metric compacta
Mathematical Subject Classification 2010
Primary: 20E08, 20F67
References
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