## EVENT DETAILS AND ABSTRACT

**Dynamical Systems Seminar**
**Title:** Quasi-isometry of nilpotent groups: an ergodic theory approach?

**Speaker:** Terence Tao

**Speaker Info:** University of California, Los Angeles

**Brief Description:**

**Special Note**:

**Abstract:**

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

**Time:** 3:00pm

**Where:** Lunt 105

**Contact Person:** Prof. Bryna Kra

**Contact email:** kra@math.northwestern.edu

**Contact Phone:** 847-491-5567

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