A family of CAT(0) outer automorphism groups of free products

Rylee Alanza Lyman

Published 2022-09-10Version 1

Consider the free product of two nontrivial finite groups with an infinite cyclic group. We prove that the 2-dimensional spine of Outer Space for this free product supports an equivariant CAT(0) metric with infinitely many ends. The outer automorphism group of this free product is thus relatively hyperbolic. In the special case that both finite groups are cyclic of order two, we show that the outer automorphism group is virtually a certain Coxeter group, and that the spine of Outer Space may be identified with its Davis-Moussong complex. These outer automorphism groups thus exhibit behavior extremely different from outer automorphism groups of free groups, and conjecturally, from other outer automorphism groups of free products of finite and cyclic groups.