The geometry of twisted conjugacy classes in wreath products

We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group $L_n$ has infinitely many twisted conjugacy classes for any automorphism $\vp$ only when $n$ is divisible by 2 or 3, originally proved by Gon\c{c}alves and Wong. We determine when the wreath product $G \wr \Z$ has this same property for several classes of finite groups $G$, including symmetric groups and some nilpotent groups.