Fall 2018

Abstract: Bestvina and Mess have shown that the group cohomology of a hyperbolic group is isomorphic to the Cech cohomology of the group’s boundary. To achieve this, they show that both cohomology groups are isomorphic to the compactly supported cohomology of the Rips complex. I will discuss recent work with Jason Manning, in which the cohomology of a relatively hyperbolic group pair is shown to be isomorphic to the Cech cohomology of the Bowditch
boundary. If there is time, I will also discuss some corollaries.

Abstract: We’ll explain how Bartholdi and Kielak’s work on cellular automata can be used to show that the integral group ring of a group satisfies the strong rank condition if and only if is amenable. This in turn can be used to prove a conjecture of Wolfgang Lueck on dimension flatness for cohomology. In addition, the results imply that the group ring can be Noetherian only if the group is amenable with max on subgroups. At present the only known such groups are polycyclic by finite.

Abstract: The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold . In particular, it predicts a 3-manifold isn’t “simple” from the perspective of Heegaard-Floer homology if and only if admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we’ll present a new theorem supporting the forward implication. Namely, we’ll discuss how to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer or foliation theories will be assumed.

Abstract: To a hyperbolic 3-manifold M, we associate the class in cohomology that computes the volume of geodesic tetrahedra in M. We will be interested in the setting that M has infinite volume, so this cohomology class is necessarily zero. To circumvent this shortcoming, we introduce bounded cohomology. As we vary the hyperbolic structure on the underlying manifold, we get potentially different bounded volume classes, and the goal of this talk will be to explain how these bounded classes change as the quasi-isometry type of the hyperbolic structure changes. Along the way, we will contemplate the classification of Kleinian groups by their end invariants and explore some interesting (bizarre) properties of bounded cohomology.

Abstract: Leighton’s graph covering theorem states that two finite graphs with isomorphic universal covers have isomorphic finite covers. I will discuss a new proof that involves using the Haar measure to solve a set of gluing equations. I will discuss generalizations to graphs with fins, and applications to quasi-isometric rigidity.

Abstract: Two far-reaching methods for studying the geometry of a finitely generated group with non-positive curvature are (1) to study the structure of the boundaries of the group, and (2) to study the structure of its finitely generated subgroups. Cannon–Thurston maps, named after foundational work of Cannon and Thurston in the setting of fibered hyperbolic 3-manifolds, allow one to combine these approaches. Mitra (Mj) generalized work of Cannon and Thurston to prove the existence of Cannon–Thurston maps for normal hyperbolic subgroups of a hyperbolic group. Such Cannon–Thurston maps can be used to obtain structure on the boundary of certain hyperbolic groups. I will explain why similar theorems fail for certain CAT(0) groups. This is joint work with Beeker–Cordes–Gardham–Gupta.

Abstract: In this talk, we will discuss the boundaries of hyperbolic groups that have enough quasiconvex codimension-1 surface subgroups. We have determined a classification for such groups when the surface subgroups have cyclic intersections. In my talk I will present the different boundaries and describe the classification. This is joint work with Nir Lazarovich.

Abstract: A celebrated result of Margulis says that among irreducible lattices in higher rank semi-simple Lie groups, arithmetic lattices are characterized as those having dense commensurators. If the subgroup of the Lie group is Zariski dense and discrete but is no longer assumed to have finite covolume (that is, to be thin), then no such definitive dichotomy exists. A heuristic due to Y. Shalom says that thin subgroups should be thought of as non-arithmetic. In this talk I will discuss a theorem confirming Shalom’s heuristic for certain naturally defined thin subgroups of . This is joint work with M. Mj.

Abstract: We study the behaviour of quasi-geodesics in Out() equipped with word metric. Given an element of Out(), there are several natural paths connecting the origin to in Out(). We show that these paths are, in general, not quasi-geodesics in Out(). In fact, we clear up the current misunderstanding about distance estimating in Out() by showing that there exists points in Out() where all quasi-geodesics between them backtracks in all of the current Out() complexes .

Abstract: This is joint work with Alexandre Martin (Heriot-Watt University). Let X be a complex of hyperbolic groups. In general the fundamental group of X need not to be hyperbolic. M. Bestvina and M. Feighn showed that if X is a graph of groups, and satisfies some natural `acylindricity’ conditions then the fundamental group of X is hyperbolic. A. Martin extended this combination theorem to the case of X whose underlying complex carries a hyperbolic CAT(0) metric. I will present a combinatorial counterpart of Martin’s result obtained recently. We introduce a weak nonpositive-curvature-like combinatorial property and show that fundamental groups of complexes of groups with underlying complex satisfying that property are hyperbolic. Our property holds for e.g. (weakly) systolic complexes and small cancellation complexes giving rise to new examples of complexes of groups with hyperbolic fundamental groups. The proof relies on constructing a potential Gromov boundary for the resulting groups and analyzing the dynamics of the action on the boundary in order to use Bowditch’s characterization of hyperbolicity.

Abstract:TBA