Geometric Property ( T )

This paper discusses ‘geometric property (T)’. This is a property of metric spaces introduced in earlier work 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, we show that geometric property (T) is a coarse invariant, i.e. depends only on the large-scale geometry of a metric space X. We also discuss how geometric property (T) interacts with amenability, property (T) for groups, and coarse geometric notions of a-T-menability. In particular, we show that property (T) for a residually finite group is characterised by geometric property (T) for its finite quotients.