Topology & Geometric Group Theory Seminar, Fall 2012
Tuesdays and some Thursdays, 1:30 – 2:30, Malott 206
Tuesday, August 28, No meeting
Tuesday, September 4, Tim Riley (Cornell)
Cannon–Thurston maps are not always well-defined
I will give an example of a hyperbolic group with a hyperbolic subgroup for which the Cannon-Thurston map is not well-defined. That is, inclusion does not induce a map of the boundaries. This is joint work with Owen Baker.
Tuesday, September 11, Collin Bleak (St Andrews)
On small presentations for the higher–dimensional Brin–Thompson groups nV
We show that all of the higher-dimensional Brin-Thompson groups nV are 2-generated, and outline work in progress towards giving small presentations of these finitely-presented, infinite simple groups.
Tuesday, September 18, Justin Moore (Cornell)
Thompson's group is amenable
Thompson's group F is a certain subgroup of the group of all piecewise linear automorphisms of ([0,1],<). I will demonstrate that this is in fact the case. This will be done by exhibiting an idempotent measure on the free nonassociative groupoid on one generator. This in turn can be used to generalize Hindman's theorem to the setting of nonassociative operations.
Justin will be giving the Logic Seminar, shortly after this seminar: 2:55pm–4:10 also in Malott 5th floor lounge. The topology seminar will focus more on background of the amenability problem for Thompson's group and will give an outline of the proof, starting with a complete proof of how amenability is related to the existence of idempotent measures. The logic seminar will briefly review the motivation for those who aren't at the topology seminar and then will focus almost exclusively on the construction of the idempotent measure. The goal will be to provide a fairly detailed and complete proof. Both talks will be fairly disjoint from eachother and either can be attended without the other without losing much.
Thursday, September 27, Moon Duchin (Tufts)
Measures on Teichmüller space
Consider the Teichmüller space T(S) of a surface S, which is the parameter space for many different kinds of geometric structures on S. T(S) itself carries a lot of structure, and accordingly has a large collection of natural measures. In joint work with Dowdall and Masur, we compare these to each other, which surprisingly goes through statements of convex geometry. We give axioms on a measure that suffice to ensure that some qualitative properties of hyperbolic geometry hold with high probability when points and rays in T(S) are sampled at random.
Thursday, October 4, Conchita Martínez-Pérez (Universidad de Zaragoza)
Isomorphisms of Brin-Higman-Thompson groups
This is a joint work with Warren Dicks. We prove that the Brin-Higman-Thompson groups sVr,n are isomorphic to certain groups of matrices over Leavitt algebras. This was observed by Pardo in the case s=1. Then using arguments available in the literature we completely determine the isomorphisms classes between the groups tVr,n (the case t=1 was also obtained by Pardo).
Tuesday, October 9, Fall Break
Tuesday, October 16
Tuesday, October 23, Boris Goldfarb (SUNY Albany)
Coarse geometry of groups and the zero divisors in their group algebras
The work of Kropholler/Linnell/Moody in the late 80's established an intimate relation between the absence of zero divisors in noetherian group algebras k[G] and the triviality of their K-theory. Their paper expressed surprise that the K-theoretic computations of Moody for torsion-free solvable groups G became possible much earlier than previously expected. Since then a lot of effort went into further successful computations for much larger classes of groups, especially many classes of geometric groups. I will outline the program of Kropholler/Linnell/Moody and show how the constraints on the group have shifted from K-theory to other parts of the argument, leading to new results for non-noetherian group algebras. A surprising aspect of this work is the importance of coarse quasi-isometry invariant properties of the group G such as having finite asymptotic dimension.
Tuesday, October 30,
Kate Juschenko (Vanderbilt) CANCELLED
We will discuss amenability of the topological full group of
a minimal Cantor system. Together with the results of H. Matui this
provides examples of finitely generated simple amenable groups. Joint
with N. Monod.
Thursday, November 1, Piotr Przytycki (Institute of Mathematics of the Polish Academy of Sciences)
Separability of embedded surfaces in 3-manifolds
This is joint work with Dani Wise. Let S be an immersed
incompressible surface in a 3-manifold M. Denote by M' the universal cover
of M. Scott proved that the group π1S is separable in π1M if and only if any
compact neighborhood of S in π1 S\M' embeds in some finite cover of M.
Rubinstein and Wang found an immersed surface which does not lift to an
embedding in a finite cover, hence violates this condition. We prove that
this is the only obstruction, i.e. that if S is already embedded, then
π1S is separable.
Tuesday, November 6, Bradley Forrest (Stockton)
A Thompson Group for the Basilica
In the 1960's, Richard J. Thompson described three groups F,
T, and V, which act by homeomorphisms on the interval, the circle, and
the Cantor set, respectively. In this talk, I will discuss joint work
with James Belk in which we define an analogous group that acts by
homeomorphisms on the Basilica Julia Set. I will also sketch our
proofs that this group is finitely generated and virtually simple.
Tuesday, November 13, Igor Rapinchuck (Yale)
On division algebras having the same maximal subfields
The talk will be built around the following question: let D1 and
D2 be two central quaternion division algebras over the same field
K; when does the fact that D1 and D2 have the same maximal
subfields imply that D1 and D2 are actually isomorphic over K?
I will discuss motivations for this question, available results, and
some generalizations to algebras of degree >2. This is joint work
with V.Chernousov and A.Rapinchuk.
Tuesday, November 20
Tuesday, November 27, Yash Lodha (Cornell)
Finiteness properties of subgroups of hyperbolic groups
Finiteness properties are important invariants of Groups. I will define the properties "type Fn", and discuss how Bestvina-Brady Morse theory can be used to establish these properties for subgroups of groups acting on CAT(0) cube complexes. I will present some new examples of subgroups of hyperbolic groups that are finitely presented but not hyperbolic.
Tuesday, December 4, Indira Chatterji (Université d'Orléans)
The median class for groups acting on CAT(0) cube complexes
I will discuss bounded cohomology, as well as CAT(0) cube complexes. For a non-elementary action on a CAT(0) cube complex, we construct a cohomology class that we call median class, and prove the non-vanishing of it. We apply this result to establish a super-rigidity result. This is joint work with T. Fernos and A. Iozzi, and this talk will be accessible to non-specialists.