---
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.'
address: |
    Institute for Algebra and Geometry\
    Karlsruhe Institute of Technology\
    Englerstr. 2\
    76128 Karlsruhe, Germany
author:
- 'Michael Schrödl-Baumann'
title: '$\ell^2$-Betti numbers of random rooted simplicial complexes'
---

introduction {#introduction .unnumbered}
============

Benjamini and Schramm introduced unimodular random rooted graphs and the notion of local weak convergence in [@benjamini-schramm], which has become a well-studied topology to investigate graphs or networks. Since in interacting complex systems not only interactions between two nodes, but also between more nodes occur, we generalize - building on a work of Elek [@elek] - a part of the extensive theory of graphs to simplicial complexes.\
A *rooted simplicial complex* is a triple $(K,V(K),x)$ consisting of a simplicial complex $K$ on a vertex set $V(K)$ and a fixed $x\in V(K)$. Usually we omit $V(K)$ and just write $(K,x)$. A simplicial complex is *locally finite* if every vertex is contained in only finitely many simplices. We say that two 