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

## TL;DR

This paper introduces a new convergence concept for finite simplicial complexes with group actions, defining $\, ext{ extltilde}\,$-homology and multiplicities that generalize $\, ext{ extltilde}\,$-Betti numbers, and studies their continuity and induction properties.

## Contribution

It develops a novel equivariant Benjamini-Schramm convergence framework for simplicial complexes with group actions, extending $\, ext{ extltilde}\,$-homology and multiplicity concepts.

## Key findings

- $\, ext{ extltilde}\,$-multiplicities are continuous on sofic random rooted complexes.
- The framework generalizes classical $\, ext{ extltilde}\,$-Betti numbers.
- Induction affects $\, ext{ extltilde}\,$-multiplicities in a predictable manner.

## 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.