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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 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.