Abstract
In this article, we contribute to the study of the dynamics by conjugation on the space of subgroups of Baumslag–Solitar groups
BS
(
m
,
n
)
$\mathrm {BS}(m,n)$
upper B upper S left parenthesis m comma n right parenthesis
via the mixing properties of elements asymptotically produced by suitable random walks on the group. In an acylindrically hyperbolic context, Hull, Minasyan and Osin [Subgroup mixing and random walks in groups acting on hyperbolic spaces.
Ergod. Th. & Dynam. Sys.
46
(10) (2026), 2505–2541] demonstrated strong mixing situations, namely topological
μ
$\mu $
mu
-mixing, a strengthening of high topological transitivity. Regarding non-metabelian
BS
(
m
,
n
)
$\mathrm {BS}(m,n)$
upper B upper S left parenthesis m comma n right parenthesis
with
|
m
|
≠
|
n
|
$|m|\neq |n|$
StartAbsoluteValue m EndAbsoluteValue not equals StartAbsoluteValue n EndAbsoluteValue
, we exhibit here a radically different situation on each of the pieces except one of the partition introduced by Carderi, Gaboriau, Le Maître and Stalder [On the space of subgroups of Baumslag–Solitar groups I: perfect kernel and phenotype.
Rev. Mat. Iberoam.
41
(5) (2025), 1711–1758] (although it is highly topologically transitive on each piece). However, when
|
m
|
=
|
n
|
$|m|= |n|$
StartAbsoluteValue m EndAbsoluteValue equals StartAbsoluteValue n EndAbsoluteValue
, we demonstrate the topological
μ
$\mu $
mu
-mixing character of the action on each of the pieces.