On invariant random subgroups of block-diagonal limits of symmetric groups

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 diagram, 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 $\chi (g) = \mathrm {Prob}(g\in K)$, where $K$ is a conjugation-invariant random subgroup of $G$.