On the one-endedness of graphs of groups

Nicholas Touikan

Vol. 278 (2015), No. 2, 463–478

DOI: 10.2140/pjm.2015.278.463

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