Expanders Learning Seminar: Expanders from property (T) groups
Speaker: Rasmus Kløvgaard Stavenuiter
Abstract: The overall theme of this lecture is to discuss the first concrete example (due to Margulis) of an expander family, arising from infinite groups with Kazhdan's property (T), which is a representation theoretical property of groups.
We begin by a brief introduction to Kazhdan's property (T) and discuss a few interesting examples, including the special linear groups SL(n, Z) , n\geq 3. (Ideas of Shalom's proof of this fact will be explained.) These groups are also known to be residually finite. We then present the proof (following Alon and Milman) that Cayley graphs of finite quotiens of property (T) residually finite infinite groups form an expander family. We also explain Gromov's result that expanders are not compatible with Euclidean structure, more precisely, that they do not "coarsely embed" into any Hilbert space.