Twisted conjugacy in free products

Daciberg Goncalves, Parameswaran Sankaran, Peter Wong

Published 2020-01-09Version 1

Let $\phi:G\to G$ be an automorphism of a group which is a free-product of finitely many groups each of which is freely indecomposable and contains a proper finite index subgroup. We show that $G$ has infinitely many $\phi$-twisted conjugacy classes. As an application, we show that if $G$ is the fundamental group of three-manifold which is not irreducible, then $G$ has the $R_\infty$-property, that is, there are infinitely many $\phi$-twisted conjugacy classes in $G$ for every automorphism $\phi$ of $G$.