Alladi Ramakrishnan Hall

Twisted conjugacy in lamplighter groups

Peter Wong

Bates College, Lewiston, USA

Given an automorphism $\varphi : G\to G$ of a group $G$, let
$R(\varphi)$ denote the
cardinality of the set of $\varphi$- twisted conjugacy classes of elements
of $G$. We say that $G$ has the property $R_{\infty}$ if $R(\varphi)=\infty$
for every $\varphi \in {\rm Aut}(G)$. The study of the finiteness of
$R(\varphi)$ stems from the classical Nielsen fixed point theory. The
so-called lamplighter groups are the wreath products $L_n=\mathbb Z_n\wr
\mathbb Z$. It is known that $L_n$ has $R_\infty$ property if and only if
$gcd(n,6)>1$. In this talk, I will give a geometric proof of this result and
discuss the analogous problem for higher rank lamplighter groups.