Reidemeister classes, wreath products and solvability

Evgenij Troitsky

Published 2023-01-29Version 1

Reidemeister (or twisted conjugacy) classes are considered in restricted wreath products of the form $G\wr \mathbb{Z}^k$, where $G$ is a finite group. For an automorphism $\varphi$ of finite order with finite Reidemeister number $R(\varphi)$, this number is identified with the number of equivalence classes of finite-dimensional unitary representations that are fixed by the dual homeomorphism $\widehat{\varphi}$ (i.e. the so-called conjecture TBFT$_f$ is proved in this case). We construct a counterexample from this class of groups to disprove the following conjecture: if a finitely generated residually finite group has an automorphism with $R(\varphi)<\infty$ then it is solvable-by-finite (so-called conjecture R).