Reidemeister numbers for arithmetic Borel subgroups in type A

Abstract

The Reidemeister number $R(\phi )$ of a group automorphism $\phi \in \mathrm{Aut}(G)$ encodes the number of orbits of the $\phi$-twisted conjugation action of $G$ on itself, and the Reidemeister spectrum of $G$ is defined as the set of Reidemeister numbers of all of its automorphisms. We obtain a sufficient criterion for some groups of triangular matrices over integral domains to have property $R_{\infty }$, which means that their Reidemeister spectrum equals $\{\infty \}$. Using this criterion, we show that Reidemeister numbers for certain soluble $S$-arithmetic groups behave differently from their linear algebraic counterparts -- contrasting with results of Steinberg, Bhunia, and Bose.

arXiv comment33 pages. v2: Substantially revised. Introduction streamlined, Theorem 1.1 reformulated. Clarifications on Levchuk's theorem added. Corrected mistakes in the proof of Proposition 5.1. This article is an improved version of the second part of arXiv:2007.02988v1, which was split into two following referee recommendations