Twisted conjugacy in soluble groups
The preprint "Twisted conjugacy in soluble arithmetic groups" by Paula Macedo Lins de Araujo and Yuri Santos Rego has been recently uploaded to the arXiv!
Link to the paper: arXiv:2007.02988.
Abstract:
We investigate the ongoing problem of classifying which S-arithmetic groups have the so-called property R∞. While non-amenable S-arithmetic groups tend to have R∞, 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 GL2 has R∞. For higher solubility class we show how automorphisms of the base ring give R∞. Our results yield many families of soluble S-arithmetic groups with R∞ but we also exhibit metabelian families not manifesting it. We formulate a conjecture concerning R∞ for the groups in question, addressing their geometric properties and algebraic structure.