Elementwise conservative actions and new constructions of boomerang subgroups

TL;DR

This paper constructs many distinct elementwise conservative non-singular random subgroups in free groups, revealing new diversity and genericity properties, and contrasting with rigidity results in higher rank lattices.

Contribution

It introduces uncountably many mutually singular elementwise conservative random subgroups in free groups, expanding understanding of subgroup structures and their generic properties.

Findings

Existence of uncountably many mutually singular such subgroups.

These subgroups are supported on infinite subgroups of infinite index.

Elementwise conservativity is generic among certain group representations.

Abstract

We show that countable non-abelian free groups admit uncountably many mutually singular elementwise conservative non-singular random subgroups, which are supported on infinite subgroups of infinite index and singular with respect to every invariant random subgroup. This complements recent rigidity results for elementwise-conservative random subgroups in higher rank lattices by the first- and third-named authors. Our proof is based on a study of representations of free groups into measurable full groups in which the action of the first generator of the free group is fixed. We show that elementwise conservativity is generic among such representations in the sense of Baire category.