UBC Mathematics Department Colloquium Hosted by PIMS-UBC 2007

A discrete group has 'Property (T)' if every isometric (affine) action of the group on Hilbert space has a global fixed point. This is a spectral-gap property and was originally defined by Kazhdan to study lattices in Lie groups. Recently Gromov formulated a probabilistic model for constructing (T) groups of very differnet character, starting with graphs with large spectral gaps ('expanders'). I shall describe the metric geometry approach to fixed point properties, and give two applications: showing that the 'wild' groups constructed by Gromov have a much stronger fixed-point property, and an algorithm which enumerates the finite presentations with property (T).