On dense totipotent free subgroups in
full groups

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Hal: https://hal.archives-ouvertes.fr/hal-02932059

arXiv : https://arxiv.org/abs/2009.03080

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}
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.