Equivariant Benjamini-Schramm Convergence of Simplicial Complexes and ${\ell }^2$-Multiplicities

Abstract

We define a variant of Benjamini-Schramm convergence for finite simplicial complexes with the action of a fixed finite group G which leads to the notion of random rooted simplicial G-complexes. For every random rooted simplicial G-complex we define a corresponding ${\ell }^2$-homology and the ${\ell }^2$-multiplicity of an irreducible representation of G in the homology. The ${\ell }^2$-multiplicities generalize the ${\ell }^2$-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study induction of random rooted complexes and discuss the effect on ${\ell }^2$-multiplicities.

20 pages, 2 figures. Comments welcome!