On dense totipotent free subgroups in full groups

TL;DR

This paper introduces totipotent ergodic actions of free groups, showing they can realize various equivalence relations and generate a continuum of distinct invariant random subgroups with full support.

Contribution

It defines totipotent actions, proves their existence for certain costs, and explores their implications for invariant random subgroups and full groups.

Findings

Every ergodic p.m.p. relation of cost < r can be realized by a totipotent free group action.

There exists a continuum of orbit inequivalent invariant random subgroups with full support.

The property of evanescence characterizes cost 1 in full groups.

Abstract

We study probability measure preserving (p.m.p.) non-free actions of free groups and the associated IRS's. The perfect kernel of a countable group Gamma is the largest closed subspace of the space of subgroups of Gamma without isolated points. We introduce the class of totipotent ergodic p.m.p. actions of Gamma: those for which almost every point-stabilizer has dense conjugacy class in the perfect kernel. Equivalently, the support of the associated IRS is as large as possible, namely it is equal to the whole perfect kernel. We prove that every ergodic p.m.p. equivalence relation R of cost $<r$ can be realized by the orbits of an action of the free group F_r on r generators that is totipotent and such that the image in the full group [R] is dense. We explain why these actions have no minimal models.This also provides a continuum of pairwise orbit inequivalent invariant random subgroups of F_r, all of whose supports are equal to the whole space of infinite index subgroups. We are led to introduce a property of topologically generating pairs for full groups (we call evanescence) and establish a genericity result about their existence. We show that their existence characterizes cost 1.