Volume 15, issue 2 (2015)

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

Masato Mimura, Narutaka Ozawa, Hiroki Sako and Yuhei Suzuki

Algebraic & Geometric Topology 15 (2015) 1067–1091
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 $k\phantom{\rule{0.3em}{0ex}}$–marked groups. In this third part of the series, we show the correspondence among the metric properties “geometric property $\left(T\right)$”, “cohomological property $\left(T\right)$” and the group property “Kazhdan’s property $\left(T\right)$”. Geometric property $\left(T\right)$ of Willett–Yu is stronger than being expander graphs. Cohomological property $\left(T\right)$ is stronger than geometric property $\left(T\right)$ for general coarse spaces.

Keywords
coarse geometry, geometric property $\mathrm{(T)}$, space of marked groups, coarse cohomology
Primary: 20F65
Secondary: 46M20