Twisted conjugacy in soluble arithmetic groups

Paula M. Lins de Araujo and Yuri Santos Rego

Mathematische Nachrichten, 2025, vol. 298, issue 3, 763-793

Abstract: 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∞$R_\infty$. 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$S$‐arithmetic groups have R∞$R_\infty$ and suggesting a conjecture in this direction.

Date: 2025
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