Texas A&M
University, Department of
Mathematics, 216 Milner Hall, 10th of September 2008, 3:00-3:50
Groups and Dynamics Seminar
Controlled coarse homology and isoperimetric inequalities
Piotr Nowak of Texas
A&M
University
We will study a controlled coarse homology theory on finitely generated
groups as a generalization of uniformly finite homology of J.Block and
S.Weinberger, and vanishing of a particular "fundamental class" in the
0th homology group. We will show that on any group one needs at most
linear control to kill the fundamental class and that this vanishing is
characterized by a certain isoperimetric inequality on the group.
Inequalities of this type were studied earlier by A.Zuk and E.Erschler.
We will also use invariants like type of asymptotic dimension,
isoperimetric profile, isodiametric profile and decay of the heat
kernel to estimate the growth necessary to kill the fundamental class.
As applications we will show a link with growth of primitives of volume
forms on open Riemannian manifolds (in particular show a short proof of
Gromov's solution to Sullivan's problem) and make a connection to
weighted Poincare inequalities studied in the context of rigidity by
P.Li and J.Wang.
This is joint work with Jan Spakula (University of Muenster), a preprint is available at http://www.math.tamu.edu/~pnowak/index/research.html.