Group approximation in Cayley topology and coarse geometry, III: Geometric property $\mathrm{(T)}$

Abstract

In this series of papers, we study the correspondence between the following: (1) the large scale structure of the metric space ${\bigsqcup }_{m}Cay\left({\mathbb{G}}^{\left(m\right)}\right)$ consisting of Cayley graphs of finite groups with $k$ generators; (2) the structure of groups that appear in the boundary of the set $\left\{{\mathbb{G}}^{\left(m\right)}\right\}$ in the space of $k$–marked groups. In this third part of the series, we show the correspondence among the metric properties “geometric property $(T)$”, “cohomological property $(T)$” and the group property “Kazhdan’s property $(T)$”. Geometric property $(T)$ of Willett–Yu is stronger than being expander graphs. Cohomological property $(T)$ is stronger than geometric property $(T)$ for general coarse spaces.