Wednesday, October 5, 2011 at 4:30pm in SM (Smith Lab) 2006
Mike Davis, Ohio State University
Right-angularity, flag complexes, asphericity
Abstract
I will discuss three related constructions of spaces and manifolds and then give necessary and sufficient conditions for the resulting spaces to be aspherical. The first construction is the "polyhedral product functor." The second construction involves applying the reflection group trick to a "corner of spaces". The third construction involves pulling back a corner of spaces via a coloring of a simplicial complex. The two main sources of examples of corners which yield aspherical results are: 1) products of aspherical manifolds with (aspherical) boundary and 2) the Borel-Serre bordifications of torsion-free arithmetic groups.
Tuesday, October 11, 2011 at 3:30pm in SM (Smith Lab) 2006
Jean-François Lafont, Ohio State University
Group theoretic decision problems: isomorphism vs. commensurability
Abstract
The isomorphism problem (IP) asks for an algorithm which inputs two finite presentations of groups, and determines whether the groups they define are isomorphic or not. The commensurability problem (CP) asks for an algorithm which determines whether the corresponding groups have finite index subgroups which are isomorphic. I'll construct a class of finitely presented groups, within which IP is unsolvable, but CP is solvable. A similar method can be used to produce a class of finitely presented groups within which CP is unsolvable, but IP is solvable. This is joint work with Goulnara Arzhantseva (Univ. Vienna) and Ashot Minasyan (U. Southampton).
Tuesday, October 18, 2011 at 3:30pm in SM (Smith Lab) 2006
Pedro Ontaneda, SUNY Binghamton
Smooth Hyperbolization
Abstract
The strict hyperbolization process of R. Charney and M. Davis produces a large and rich class of negatively curved spaces (in the geodesic sense). This process is based on an earlier version introduced by M. Gromov and later studied by M. Davis and T. Januszkiewicz. If M is a manifold its Charney-Davis strict hyperbolization h(M) is also a manifold, but the negatively curved metric obtained is far from being Riemannian because it has a large and complicated set of singularities. We will discuss whether this process can be done smoothly.
Tuesday, November 15, 2011 at 3:30pm in SM (Smith Lab) 2006
Robin Lassonde, University of Michigan
Splittings of non-finitely generated groups
Abstract
A splitting of a group G is an algebraic generalization of a codimension-1 submanifold of a manifold whose fundamental group is G. One can view a splitting as a G-action on a tree. In 1998, P. Scott de ned the intersection number of two splittings (or, more generally, two almost invariant subsets) of a nitely generated group. Over the next several years, the theory of intersection number of almost invariant sets was developed by G. Niblo, M. Sageev, P. Scott and G. Swarup. They required the ambient group G to be nitely generated, and usually also required the associated subgroups to be nitely generated. I rework the theory to remove both nite generation assumptions, in the case when the almost invariant sets arise from splittings. Whereas the aforementioned authors used the Cayley graph of G, I capitalize on the geometry of trees. In this talk, I will give motivating examples and present some of the main results.
Tuesday, December 13, 2011 at 2:30pm in MW 154
Thomas Koberda, Harvard University
Right-angled Artin subgroups of right-angled Artin groups
Abstract
I will present a systematic way of classifying all right-angled Artin subgroups of a given right-angled Artin group. The methods used have a number of corollaries: for instance, it can be shown that there is an embedding between a right-angled Artin group on a cycle of length m to one on a cycle of length n if and only if m=n+k(n-4) for some nonngeative integer k. I will also give some rigidity results. This is joint work with Sang-hyun Kim.