On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype

TL;DR

This paper investigates the topological and dynamical structure of the subgroup space of Baumslag-Solitar groups, revealing a natural partition, the concept of phenotype, and properties like topological transitivity within each partition.

Contribution

It introduces the notion of phenotype to describe subgroup partitions and analyzes the topological dynamics of the subgroup space, including perfect kernel and invariant subspaces.

Findings

Identified the perfect kernel of the subgroup space.

Partitioned the subgroup space into invariant subsets with transitive actions.

Defined the phenotype function to classify subgroup partitions.

Abstract

Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural partition of the space of subgroups into one closed subset and countably many open subsets that are invariant under the action by conjugation. One of our main results is that the restriction of the action to each piece is topologically transitive. This partition is described by an arithmetically defined function, that we call the phenotype, with values in the positive integers or infinity. We eventually study the closure of each open piece and also the closure of their union. We moreover identify in each phenotype a (the) maximal compact invariant subspace.