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

Abstract:

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.