On Invariant Random Subgroups of Block-Diagonal Limits of Symmetric Groups

TL;DR

This paper classifies ergodic invariant random subgroups of certain block-diagonal limits of symmetric groups, linking them to stabilizer distributions and group characters, under conditions of simplicity and finite-dimensionality.

Contribution

It provides a complete classification of ergodic invariant random subgroups for simple full groups associated with Bratteli diagrams with finite ergodic measures.

Findings

Non-Dirac ergodic invariant random subgroups are stabilizer distributions on finite products.

Group characters are expressed as probabilities of membership in conjugation-invariant random subgroups.

Every such group character corresponds to a stabilizer distribution of a conjugation-invariant subgroup.

Abstract

We classify the ergodic invariant random subgroups of block-diagonal limits of symmetric groups in the cases when the groups are simple and the associated dimension groups have finite dimensional state spaces. These block-diagonal limits arise as the transformation groups (full groups) of Bratteli diagrams that preserve the cofinality of infinite paths in the diagram. Given a simple full group $G$ admitting only a finite number of ergodic measures on the path-space $X$ of the associated Bratteli digram, we prove that every non-Dirac ergodic invariant random subgroup of $G$ arises as the stabilizer distribution of the diagonal action on $X^n$ for some $n\geq 1$. As a corollary, we establish that every group character $\chi$ of $G$ has the form $\chi(g) = Prob(g\in K)$, where $K$ is a conjugation-invariant random subgroup of $G$.