\(\ell^2\)-Betti numbers of random rooted simplicial complexes (Q2188924)

In the paper under review, the author defines unimodular measures on the space of rooted simplicial complexes, and he associates to each such measure a chain complex and a trace function. This is based on work of \textit{I. Benjamini} and \textit{O. Schramm} [Electron. J. Probab. 6, Paper No. 23, 13 p. (2001; Zbl 1010.82021)] in which these authors introduce the notions of unimodular random rooted graph and of local weak convergence which give rise to a topology on graphs or networks. As a consequence, the author defines 2-Betti numbers of unimodular random rooted simplicial complexes and shows that they are continuous under Benjamini-Schramm convergence. Furthermore, he gives a characterization of the limit of normalized Betti numbers of a Benjamini-Schramm convergent sequence of finite simplicial complexes of uniformly bounded degree.