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than geometric property (T) for general coarse spaces.Comment: 20 pages, Appendix withdrawn due to the error in the proof of Theorem
A.2 (v3); 24 pages, Appendix added (v2); 20 pages, no figur
