Twisted conjugacy in residually finite groups of finite Prüfer rank

Abstract

Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $\phi$ with finite Reidemeister number $R(\phi )$ (the number of $\phi$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_{\infty }$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $\phi$ is its automorphism with $R(\phi )<\infty$; then $R(\phi )$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations of $G$, which are fixed points of the dual map $\hat{\phi }:[\rho ]\mapsto [\rho \circ \phi ]$ (i.e., we prove the TBFT${}_f$, the finite version of the conjecture about the twisted Burnside-Frobenius theorem, for such groups).

arXiv comment9 pages