Twisted conjugacy in soluble arithmetic groups

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here, we address a question posed by Gonçalves and Wong in the mid-2000s: we construct an infinite series of compact connected solvmanifolds (that are not nilmanifolds) of strictly increasing dimensions and all of whose self-homotopy equivalences have vanishing Nielsen number. To this end, we establish a sufficient condition for a prominent (infinite) family of soluble linear groups to have the so-called property $R_{\infty }$. In particular, we generalize or complement earlier results due to Dekimpe, Gonçalves, Kochloukova, Nasybullov, Taback, Tertooy, Van den Bussche, and Wong, showing that many soluble $S$-arithmetic groups have $R_{\infty }$ and suggesting a conjecture in this direction.