Twisted conjugacy in free products

Abstract

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

7 pages