Subgroup mixing in Baumslag-Solitar groups

TL;DR

This paper investigates the dynamics of subgroup conjugation actions in Baumslag-Solitar groups, revealing diverse mixing behaviors depending on group properties, with implications for understanding their complex topological dynamics.

Contribution

It provides new insights into the mixing properties of conjugation actions in Baumslag-Solitar groups, especially distinguishing between unimodular and non-unimodular cases.

Findings

Non-unimodular BS(m,n) exhibit different mixing behaviors on each partition piece.

Unimodular BS(m,n) demonstrate topological mu-mixing on all pieces.

Strong mixing results extend previous work in acylindrically hyperbolic contexts.

Abstract

In this article, we contribute to the study of the dynamics induced by the conjugation action on the space of subgroups of Baumslag-Solitar groups BS(m,n), via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, the authors of [HMO] demonstrated strong mixing situations, namely topological mu-mixing, a strengthening of high topological transitivity. Regarding non-metabelian and non-unimodular BS(m,n), we exhibit here a radically different situation on each of the pieces except one of the partition introduced in [CGLMS22] (although it is highly topologically transitive on each piece). On the other hand, when BS(m,n) is unimodular, we demonstrate the topological mu-mixing character on each of the pieces.