$\ell^2$-Betti numbers of random rooted simplicial complexes

TL;DR

This paper introduces a new framework for defining and analyzing $\,\ell^2$-Betti numbers of unimodular random rooted simplicial complexes, demonstrating their continuity under Benjamini-Schramm convergence.

Contribution

It develops a novel approach to assign $\,\ell^2$-Betti numbers to unimodular measures on rooted simplicial complexes, linking measure theory with algebraic topology.

Findings

$\,\ell^2$-Betti numbers are well-defined for unimodular random rooted simplicial complexes.

These Betti numbers are continuous with respect to Benjamini-Schramm convergence.

The framework connects measure-theoretic and topological properties of random complexes.

Abstract

We define unimodular measures on the space of rooted simplicial complexes and associate to each measure a chain complex and a trace function. As a consequence, we can define $\ell^2$-Betti numbers of unimodular random rooted simplicial complexes and show that they are continuous under Benjamini-Schramm convergence.