Next talk:

(organizers: Mikael Pichot and Dani Wise )

Location: The seminar meets on Wednesday at 3pm in BURN920

September 5th, 2012

Dismantlability

Abstract: I will explain what it means for a finite graph to be "dismantlable" and

sketch a proof of Polat's theorem that such a graph has a clique fixed

under all of its automorphisms. I will present applications to various

infinite graphs appearing in geometric topology. The latter is joint work

with Hensel and Osajda.

Piotr Przytyki

Previous talks:

September 12th, 2012

A problem about generic curves in surfaces

Abstract: Let S be a finite type hyperbolic surface.

My aim is to formulate a problem about the behavior of generic closed geodesics in S. I am interested in the problem because it is connected to basic issues in combinatorial group theory. As will be explained, a positive resolution of the problem has applications towards the coherence and local quasiconvexity of one-relator groups.

Dani Wise

May 15

Random walks, boundary values and vanishing of l^p cohomology

Given a graph (of bounded valency), there is a natural operation from function on vertices to function on edges given by the gradient (the difference at the extremities of the edge). Simply put, the l^p cohomology (in degree 1) of this graph is the quotient of the space of functions with gradient in l^p(E) by the space of functions in l^p(X). When the graph is the Cayley graph of a group G, this coincides with the cohomology of the left-regular representation of G in l^p(G). These quotient are also useful invariants (of quasi-isometry).

In this talk, a natural map from reduced l^p cohomology to harmonic functions (under relatively mild growth assumptions on the graph) will be constructed. Among the consequences, a partial answer to a question of Pansu for the will be given, namely, for the Cayley graph of a group of superpolynomial growth: if there are no non-constant bounded harmonic functions whose gradient is l^p, then the reduced l^q cohomolgy is trivial for all p<q.

This allows to make significant progress on a question of Gromov (whether the reduced l^p cohomology for all amenable groups and all 1<p<\infty vanishes or not). First, one gets that the reduced l^p cohomology is trivial for all 1<p<2 and all amenable groups. Second, if the group is Liouville, then the reduced l^p cohomology is trivial for all 1<p<\infty.

Antoine Gournay

March 6

Relative hyperbolicity and thickness of cubulated groups

Abstract: This talk is on joint work with Jason Behrstock. I will discuss the simplicial boundary of a CAT(0) cube complex, which is an analogue of the Tits boundary defined in terms of the combinatorial geometry (without reference to CAT(0) geometry). The simplicial boundary provides a means of identifying relative hyperbolicity, and its "opposite" -- thickness -- of groups acting properly and cocompactly on CAT(0) cube complexes: relative hyperbolicity and thickness are both characterized by simple topological conditions on the action of the group on the simplicial boundary. I will make this statement precise and discuss some related open questions.

Mark Hagen

March 27

Deformation of algebras from group 2-cocycles

Algebras with graded by a discrete can be deformed using 2-cocycles on the base group. We give a K-theoretic isomorphism of such deformations, generalizing the previously known cases of the theta-deformations and the reduced twisted group algebras. When we perturb the deformation parameter, the monodromy of the Gauss-Manin connection can be identified with the action of the group cohomology.

Makoto Yamashita