Hierarchies and semistability of relatively hyperbolic groups

G. Christopher Hruska, Kim Ruane

Published 2019-04-29Version 1

A finitely presented group is semistable if all proper rays in the Cayley 2-complex are properly homotopic. A long standing open question asks whether all finitely presented groups are semistable. In this article, we prove semistability of groups that are hyperbolic relative to polycyclic subgroups. Key tools in the proof are a result of Mihalik-Swenson on semistability of `atomic' relatively hyperbolic groups, a combination theorem of Mihalik-Tschantz, and a hierarchical accessibility theorem of Louder-Touikan. We analyze an example that illustrates why an understanding of hierarchies is necessary for the proof of semistability in this context.