Isoperimetric inequalities and random walks on quotients of graphs and buildings (Q1882591)

Certain group-theoretic properties of locally compact groups are reflected by Kazhdan's property. The paper under review deals with the study of connected, locally finite graphs \({\mathcal G}\), whose vertex sets can be written as homogeneous spaces \(G/B\), where \(B\subset G\subset \text{Aut\,} {\mathcal G}\) is open and compact in \(G\). The author proves that any graph \({\mathcal G}\), such that \(G\) has Kazhdan's property, and all its quotients by discrete subgroups \(\Gamma\subset G\) satisfy strong isoperimetric inequalities. Moreover, uniform positive lower bounds for the Cheeger constants are obtained. Such graphs cannot be bi-Lipschitz embedded in Hilbert spaces.