  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 of finite graphs $(X_n)$, it is strictly stronger
than $(X_n)$ being an expander in the sense that the Cheeger constants $h(X_n)$
are bounded below.
  We show here 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 the
relationships between geometric property (T) and amenability, property (T), and
various 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.
