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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

In this series of papers, we study the correspondence between the following: (1) the large scale structure of the metric space m Cay(G(m)) consisting of Cayley graphs of finite groups with k generators; (2) the structure of groups that appear in the boundary of the set {G(m)} 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.

coarse geometry, geometric property $\mathrm{(T)}$, space of marked groups, coarse cohomology
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 46M20
Received: 21 May 2014
Accepted: 26 August 2014
Published: 22 April 2015
Masato Mimura
Mathematical Institute
Tohoku University
Sendai 980-8578
Narutaka Ozawa
Research Institute for Mathematical Sciences
Kyoto University
Kyoto 606-8502
Hiroki Sako
School of Science
Tokai University
Hiratsuka 259-1292
Yuhei Suzuki
Department of Mathematical Sciences
University of Tokyo
Tokyo 153-0041