Amenability, cohomology and property A

Graham Niblo (University of Southampton)

Wednesday 25th November, 2009 15:00-16:00 Mathematics Building, Room 516


* Amenability appears as one of the fundamental concepts bridging the worlds of functional analysis and geometric group theory. A group is said to be amenable if it admits an invariant mean on the space of bounded functions on the group. While the definition can be extended to an abstract metric space using Folner's criterion instead, in the absence of a group action the notion is not sufficiently powerful to encode the coarse geometry of the space and this is not a particularly fruitful approach. In his work on the Novikov conjecture Yu introduced an alternative non-equivariant generalisation of amenability, Yu's Property A, in which equivariance is replaced by a controlled support condition which captures more of the geometry. Spaces satisfying Yu's condition also satisfy the Coarse Baum Connes conjecture. There are several well known homological characterisations of amenability and Higson asked if there are analogous characterisations of property A. We will consider coarse generalisations of bounded cohomology and Block & Weinberger's uniformly finite homology which provide a positive answer to Higson's question and illuminate the extent to which property A provides asymptotic means on a group.

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