Twisted conjugacy in soluble arithmetic groups

TL;DR

This paper investigates twisted conjugacy in soluble arithmetic groups, constructing specific solvmanifolds with vanishing Nielsen numbers and establishing conditions for groups to have the property R_infinity, advancing understanding of group automorphisms.

Contribution

It constructs an infinite series of solvmanifolds with vanishing Nielsen numbers and provides a sufficient condition for soluble linear groups to have the R_infinity property, extending prior results.

Findings

Constructed solvmanifolds with all self-homotopy equivalences having zero Nielsen number.

Established a condition for soluble linear groups to possess the R_infinity property.

Generalized previous results on soluble S-arithmetic groups and proposed a related conjecture.

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 Gon\c{c}alves 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, Gon\c{c}alves, 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.