Twisted conjugacy in soluble arithmetic groups

Abstract

We investigate the ongoing problem of classifying which S-arithmetic groups have the so-called property $R_\infty$. While non-amenable S-arithmetic groups tend to have $R_\infty$, the soluble case seems more delicate. Here we address Borel subgroups in type A and show how the problem reduces to determining whether a metabelian subgroup of $\mathrm{GL}_2$ has $R_\infty$. For higher solubility class we show how automorphisms of the base ring give $R_\infty$. Our results yield many families of soluble S-arithmetic groups with $R_\infty$ but we also exhibit metabelian families not manifesting it. We formulate a conjecture concerning $R_\infty$ for the groups in question, addressing their geometric properties and algebraic structure.Comment: 47 page