Reidemeister numbers for arithmetic Borel subgroups in type A

The Reidemeister number $R(\phi )$ of a group automorphism $\phi \in \mathrm{A}\mathrm{u}\mathrm{t}(G)$ 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 $R_{\mathrm{\infty }}$, which means that their Reidemeister spectrum equals $\{\mathrm{\infty }\}$. 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.