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This article is available for purchase or by subscription. See below.
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.
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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