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

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.