Geometric property (T)

Abstract

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 (X _n) of bounded degree finite graphs, it is strictly stronger than (X _n) being an expander in the sense that the Cheeger constants h(X _n) 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.

Keywords

Coarse geometry / Expander / Roe algebra / Property (T)

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