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

Abstract

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

20 pages, Appendix withdrawn due to the error in the proof of Theorem A.2 (v3); 24 pages, Appendix added (v2); 20 pages, no figure