# Geometric Analysis and Topology Seminar

#### Expanders with no Uniform Embedding in a Uniformly Convex Banach Space

Speaker: Vincent Lafforgue, Jussieu

Location: Warren Weaver Hall 1013

Date: Friday, February 29, 2008, 1 p.m.

Synopsis:

We show that if $$F$$ is a local non-archimedian field the trivial representation of $$S L_3(F)$$ is isolated among isometric representations in uniformly convex Banach spaces and deduce from this strong form of property $$(T)$$ that the expanders constructed as finite quotients of a lattice in $$S L_3(F)$$ do not embed uniformly in a uniformly convex Banach space. The same question for Ramanujan expanders associated to quaternions is open: a strong form of property (tau) would be needed. Notes from a previous talk on this subject are available at the address http://www.institut.math.jussieu.fr/~vlafforg/edimbourg.pdf.