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 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 $n\ge 1$. 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 $G$.

14 pages, 1 figure

Cited by in corpus (1)

• On rigid stabilizers and invariant random subgroups of groups of homeomorphisms