Subgroup mixing in Baumslag-Solitar groups

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.

Notation 2.3 and Notation 2.9 added. More details added to Remark 2.2, Remark 2.4, and Remark 2.10. A strategy of the proof added at the beginning of Section 3.2. Lemma 3.4 and Lemma 3.6 rephrased in terms of the action on the tree. Proofs of Lemma 3.8 and Lemma 3.10 compactified. Some details added to the proof of Proposition 3.6. Some typos corrected