Title: Quasi-isometry of nilpotent groups: an ergodic theory approach?
Speaker: Terence Tao
Speaker Info: University of California, Los Angeles
Roughly speaking, two metric spaces are said to be quasi-isometric if there is a map from one space to the other which is bilipschitz at large scales, and approximately surjective; thus, for instance, Z^n and R^n are quasi-isometric. A well-known open problem in geometric group theory is to understand which finitely generated nilpotent (or virtually nilpotent) groups (or equivalently, the groups of polynomial growth) are quasi-isometric to each other; certainly any two commensurate such groups are quasi-isometric, but it is not known if the converse is true. There are two families of quasi-isometric invariants for nilpotent groups known. The first, due to Pansu, asserts that the Carnot groups of quasi-isometric nilpotent groups are isomorphic; roughly speaking, this means that quasi-isometric groups are isomorphic "to top order". The second, due to Shalom and refined by Sauer, shows that certain higher order group cohomological data are also quasi-isometric invariants. The two sets of invariants are not directly comparable, but interestingly they can both be derived by ergodic-theory methods, raising the possibility (which, unfortunately, is so far unrealised) that more advanced methods in ergodic theory might yield further progress on this problem.Date: Tuesday, May 01, 2012