Equivariant Benjamini–Schramm convergence of simplicial complexes and ℓ2-multiplicities
Steffen Kionke, Michael Schrödl-Baumann · Journal of Topology and Analysis · 2020
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 unimodular random rooted simplicial G-complexes. For every unimodular random rooted simplicial G-complex we define a corresponding [Formula: see text]-homology and the [Formula: see text]-multiplicity of an irreducible representation of G in the homology. The [Formula: see text]-multiplicities generalize the [Formula: see text]-Betti numbers and we show that they are continuous on the space of sofic random rooted simplicial G-complexes. In addition, we study the induction of random rooted complexes and discuss the effect on [Formula: see text]-multiplicities.