###### 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 dened 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.