Montreal Geometric & Combinatorial Group Theory Seminar




Speaker: Iosif Polterovich (U Montreal)

Title: “Trees, groups and asymptotic cones

Date: 3:30pm Wednesday, November 27.

Room: 920 Burnside




We discuss some results on real trees and asymptotic cones of groups. Asymptotic cones proved to be a very useful tool in geometric group theory. For example, a celebrated theorem of Gromov on groups of polynomial growth was proved with the help of asymptotic cones. As was shown also by Gromov, any asymptotic cone of a hyperbolic metric space is a real tree. In a joint work with A. Dyubina we suggested some explicit constructions of real trees, appearing in the asymptotic geometry of hyperbolic spaces. In particular, we have shown that asymptotic cones of hyperbolic manifolds and groups are independent of the choice of an ultrafilter. Note that if we do not assume that the group is hyperbolic the result is not true: there exists a finitely generated group with non-homeomorphic asymptotic cones. This observation is due to S. Thomas and B. Velickovic and we also discuss it in our talk.