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 3-manifold groups.

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 Tadeusz Januszkiewicz.)

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.

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 3-manifold groups.

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 Tadeusz Januszkiewicz.)

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.