Group approximation in Cayley topology and coarse geometry, III: Geometric property (T)

Mimura, Masato¹ ; Ozawa, Narutaka² ; Sako, Hiroki³ ; Suzuki, Yuhei⁴

¹Mathematical Institute, Tohoku University, Sendai 980-8578, Japan
²Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
³School of Science, Tokai University, Hiratsuka 259-1292, Japan
⁴Department of Mathematical Sciences, University of Tokyo, Tokyo 153-0041, Japan

Algebraic and Geometric Topology, Tome 15 (2015) no. 2, pp. 1067-1091

Résumé

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.

Détail

DOI : 10.2140/agt.2015.15.1067

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

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