Geometric Property(T)

This paper discusses "geometric property(T)". This is a property of metric spaces introduced in earlier works of the authors for its applications to K-theory. Geometric property(T) is a strong form of "expansion property", in particular, for a sequence(Xn)of bounded degree finite graphs, it is strictly stronger than(Xn) being an expander in the sense that the Cheeger constants h(Xn) are bounded below.In this paper, the authors show that geometric property(T) is a coarse invariant,i.e., it depends only on the large-scale geometry of a metric space X. The authors also discuss how geometric property(T) interacts with amenability, property(T) for groups,and coarse geometric notions of a-T-menability. In particular, it is shown that property(T) for a residually finite group is characterised by geometric property(T) for its finite quotients.