37 research outputs found
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 , it is strictly stronger
than being an expander in the sense that the Cheeger constants
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 . 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.Comment: Version two corrects some typos and a mistake in the proof of Lemma
8.
A metric approach to limit operators
We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from Z^N to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property