Group theory seminar: 2012-2013 calendar

Organizers: Mikael Pichot, Dani Wise

The seminar meets on Wednesday at 3pm in BURN920.

Random walks, boundary values and vanishing of $\ell^p$ cohomology

Antoine Gournay (Neuchatel)
May 15, 2013, 3pm
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 $\ell^p$ cohomology (in degree 1) of this graph is the quotient of the space of functions with gradient in $\ell^p(E)$ by the space of functions in $\ell^p(X)$. When the graph is the Cayley graph of a group G, this coincides with the cohomology of the left-regular representation of G in $\ell^p(G)$. These quotient are also useful invariants (of quasi-isometry).
In this talk, a natural map from reduced $\ell^p$ 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 $\ell^p$, then the reduced $\ell^q$ cohomolgy is trivial for all $p < q$.
This allows to make significant progress on a question of Gromov (whether the reduced $\ell^p$ cohomology for all amenable groups and all $1 < p < \infty$ vanishes or not). First, one gets that the reduced $\ell^p$ cohomology is trivial for all $1 < p < 2$ and all amenable groups. Second, if the group is Liouville, then the reduced $\ell^p$ cohomology is trivial for all $1 < p < \infty$.

Dixmier's similarity problem

Narutaka Ozawa (RIMS, Kyoto)
April 17, 2013, at 3pm
A group G is said to be unitarizable if every uniformly bounded representation of G on a Hilbert space is similar to a unitary representation Sz.-Nagy, Dixmier and Day proved that amenability implies unitarizability, and Dixmier posed a problem whether the converse is also true. I will report on not so recent anymore progress on Dixmier's Similarity Problem. This is a joint work with N. Monod.

Deformation of algebras from group 2-cocycles

Makoto Yamashita (Tokyo)
March 27, 2013, at 3pm
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.

Relative hyperbolicity and thickness of cubulated groups

Mark Hagen
March 6, 2013, at 3pm
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.


Piotr Przytyki
September 5th, 2012, at 3pm
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.

A problem about generic curves in surfaces

Dani Wise
September 12th, 2012, at 3pm
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.