One-endedness of outer automorphism groups of free products of finite and cyclic groups

Rylee Alanza Lyman

Published 2023-05-08Version 1

In a previous paper, we showed that the group of outer automorphisms of the free product of two nontrivial finite groups with an infinite cyclic group has infinitely many ends, despite being of virtual cohomological dimension two. The main result of this paper is that aside from this exception, having virtual cohomological dimension at least two implies the outer automorphism group of a free product of finite and cyclic groups is one ended. As a corollary, the outer automorphism group of the free product of four finite groups or the free product of a single finite group with a free group of rank two is a virtual duality group of dimension two, in contrast with the above example. We also prove that groups in this family are semistable at infinity (or at each end). Our proof is inspired by methods of Vogtmann, applied to a complex first studied in another guise by Krsti\'c and Vogtmann.