Vol. 278, No. 2, 2015

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On the one-endedness of graphs of groups

Nicholas Touikan

Vol. 278 (2015), No. 2, 463–478
Abstract

We give a technical result that implies a straightforward necessary and sufficient condition for a graph of groups with virtually cyclic edge groups to be one-ended. For arbitrary graphs of groups, we show that if their fundamental group is not one-ended, then we can blow up vertex groups to graphs of groups with simpler vertex and edge groups. As an application, we generalize a theorem of Swarup to decompositions of virtually free groups.

Keywords
group theory, groups acting on trees, Bass–Serre theory, ends of groups, square complexes
Mathematical Subject Classification 2010
Primary: 20E06, 20E08
Secondary: 57M07, 57M60, 57M20
Milestones
Received: 2 September 2014
Revised: 25 March 2015
Accepted: 25 March 2015
Published: 6 October 2015
Authors
Nicholas Touikan
Department of Mathematical Sciences
Stevens Institute of Technology
1 Castle Point Terrace
Hoboken, NJ 07030
United States