Title and Abstracts: WTC 2009
Dani Wise, McGill
Title: Groups with a quasiconvex hierarchy
Abstract: Nonpositively curved cube complexes have become increasingly central objects in combinatorial and geometric group theory. We will give a quick introduction focusing especially on their identity as "higher dimensional graphs". We will then outline how to use nonpositively curved cube complexes to resolve certain outstanding group theoretical problems. These include Baumslag's conjecture on the residual finiteness of one-relator groups with torsion, as well as the virtual-fibering problem for Haken hyperbolic 3-manifolds.
Denis Osin, Vanderbilt
Title: Hyperbolically embedded subgroups in groups acting on hyperbolic spaces
Abstract: The aim of this talk is to propose a generalization of
relative hyperbolicity based on the notion of a hyperbolically
embedded subgroup. On the one hand, many results about peripheral
subgroups of relatively hyperbolic groups can be naturally generalized
to hyperbolically embedded subgroups. On the other hand, many groups
which in general do not admit any nontrivial relatively hyperbolic
structure (e.g., mapping class groups, outer automorphism groups of
free groups, groups acting on trees, etc.) do contain nontrivial
hyperbolically embedded subgroups. This allows us to obtain some new
results about these groups.
Thomas Koberda, Harvard
Title: Representations of mapping class groups and residual properties of
Abstract: I will talk about homological representations of mapping class groups,
namely ones which arise from actions on covering spaces. I will prove
that these are asymptotically faithful and indicate how the
Nielsen-Thurston classification can be obtained from these
representations. I will then discuss how mapping tori of mapping classes
can be used to analyze the image of these representations. As a
corollary, I will exhibit a class of compact 3-manifolds whose fundamental
groups are, for every prime p, virtually residually finite p.
Pallavi Dani, LSU
Title: Higher divergence for right-angled Artin groups
Abstract: The theory of higher dimensional Dehn functions asks extremal questions for filling k-spheres with (k+1)-balls in complexes associated to the group. Topology at infinity is the study of the asymptotic structure of groups by attaching topological invariants to the complements of large balls. Higher divergence functions bring these two ideas together, by studying rates of filling "at infinity" in groups and other metric spaces. I will talk about the basic definitions and motivation for these functions, followed by recent results on higher divergence in the class of right-angled Artin groups. This is joint work with A. Abrams, N. Brady, M. Duchin, A. Thomas and R. Young.
Joan Licata, Stanford
Title: Invariants for Legendrian knots in contact lens spaces
Abstract: Relative contact homology is a Floer-theoretic invariant for Legendrian knots in contact manifolds, and in special cases, it can be computed combinatorially from the Lagrangian projection of the knot. In this talk, I'll describe these combinatorial invariants, focusing on the case of primitive knots in lens spaces.
Amir Mohammadi, Chicago
Title: Discrete vertex transitive action on Bruhat-Tits building.
Abstract: In this talk we will report on a joint work with A. Salehi Golsefidy in
which the aim was to classify all maximal transitive actions on higher
rank algebraic Bruhat-Tits buildings over local fields of characteristic
zero. Using Tits' classification of the isometry group of the building the
problem will reduce to type A. Unlike the action on the trees, where there
are many transitive actions, we will see that such actions are very rare
in here. A corollary of our work is that there is no such action on A_n
for n bigger than 7.
Jason Behrstock, CUNY
Title: Quasi-isometric classification of right angled Artin groups
Abstract: Any finitely generated group can be endowed with a natural
metric which is unique up to maps of bounded distortion
(quasi-isometries). A fundamental question is to classify finitely
generated groups up to quasi-isometry. Surprisingly, for a large
family of right angled Artin groups the quasi-isometric classification
can be described in terms of a concept in computer science called
"bisimulation." We will describe this classification and a geometric
interpretation of bisimulation. (Joint work with Walter Neumann and
Johanna Mangahas, Michigan
Title: "Short-word" pseudo-Anosovs
Abstract: The problem of constructing "short-word" pseudo-Anosovs relates to proving effective versions of the Tits alternative for the mapping class group. I'll describe both problems, their relation, and their solutions.