Schedule and Abstracts

The notion of "filling" of groups has been very fruitful over recent years.  Motivated by questions around the Cannon Conjecture, I will explain how to take a (residually finite) hyperbolic group with a two-sphere boundary and "drill" a finite-index subgroup to produce a relatively hyperbolic group with two-sphere boundary.  This allows us to relate the Cannon Conjecture to the relatively hyperbolic version.

This is joint work with Peter Haïssinsky, Jason Manning, Damian Osajda, Alessandro Sisto, and Genevieve Walsh.

Artin groups, which will be introduced by Ruth in her minicourse, are a family of groups generalizing braid groups and are closely related to Coxeter groups. There is a conjectural description of the center of every Artin group. Irreducible Artin groups (i.e. those that do not split as direct products) are conjectured to have trivial centers, unless they are of finite type, in which case they are known to have infinite cyclic centers. In my talk, I will present joint work with Kevin Schreve, where we show that the Artin groups satisfying the K(pi,1) conjecture also satisfy the center conjecture. This uses representations of Artin groups in mapping class groups, due to Crisp-Paris, which I will also discuss.

An important technique in the theory of 3-manifolds is to cut a 3-manifold up along tori to obtain a JSJ decomposition of the 3-manifold. There is a group-theoretic analogue, due to Rips-Sela, in which one studies all ways in which a finitely presented group can be split as an amalgamated free product or HNN extension along a specified family of subgroups. More generally, one can aim to prove an "algebraic torus theorem" in the sense of Dunwoody-Swenson: a structure theorem for codimension-one subgroups of a group within some family of subgroups. This is a group-theoretic analogue of studying a 3-manifold by investigating its immersed tori.

In this talk, we use cohomological methods to investigate the structure of group splittings of a hyperbolic group over quasi-convex Poincare duality groups that are not necessarily slender or small. We prove a JSJ theorem and an algebraic torus theorem for hyperbolic groups that are acyclic at infinity.

Free-by-cyclic groups can be defined as mapping tori of free group automorphisms. I will discuss various dynamical properties of automorphisms that turn out to be group invariants of the corresponding free-by-cyclic groups (e.g. growth type). In particular, certain dynamical hierarchical decompositions of an automorphism determine canonical hierarchical decompositions of its mapping torus.

With every group G, one can naturally associate its von Neumann algebra, denoted by L(G). If G is amenable, L(G) usually admits a huge (uncountable) group of outer automorphisms that contains all countable groups. On the other hand, A. Connes proved that Out(L(G)) is countable whenever G has Kazhdan's property (T). In 2000, V.F.R. Jones proposed a conjecture predicting the precise structure of Out(L(G)) for such groups. Despite significant recent progress in understanding group von Neumann algebras, the conjecture remained wide open: neither a counterexample nor a single non-trivial example of a group satisfying it has been found thus far. In my talk, I will discuss some results of my recent joint work with A. Ioana, I. Chifan, and B. Sun motivated by Conne's theorem and Jones' conjecture.  In particular, we prove  Jones' conjecture for a wide class of groups that occur in the context of group theoretic Dehn filling. As an application, we show that every countable group realizes as Out(L(G)) for some countable group G with property (T). 

Thurston proved that a positive real number is the topological entropy for an ergodic traintrack representative of an outer automorphism of a free group if and only if its expansion constant is a weak Perron number. This is a powerful result, answering a question analogous to one regarding surfaces and stretch factors of pseudo-Anosov homeomorphisms. However, Thurston fell ill during the writing of his paper and his proof is difficult to parse. In joint work with a group of graduate students at the University of Utah, we modernized Thurston's approach, filled in gaps in the original paper, and distilled Thurston's methods to give a cohesive proof of the traintrack theorem, which I will present in this talk. 

TBA

The Poisson boundary is a measure-theoretic object attached to a group  equipped with a probability measure, and is closely related to the notion of harmonic function on the group. In many cases, the group is also endowed with a topological boundary arising from its geometric structure, and a recurring research theme is to discuss the relation between the two notions of boundary. 

In this talk, we prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. In this generality, this identification result is new even for free groups. We will then discuss extensions of this result to other groups with hyperbolic properties. 

Joint with K. Chawla, B. Forghani, and J. Frisch. 

The 1973 Boone-Higman Conjecture predicts that every finitely generated group with solvable word problem embeds in a finitely presented simple group. In joint work with Jim Belk, Collin Bleak, and Francesco Matucci, we prove that all hyperbolic groups satisfy the Boone-Higman Conjecture, that is, every hyperbolic group embeds into some finitely presented simple group. The simple groups in question are certain so-called twisted Brin-Thompson groups, introduced by Belk and myself a few years ago. In order to obtain our embeddings, we use a middleman: a new family of "contracting" groups V_T[N] of homeomorphisms of Cantor space that are interesting in their own right. In this talk I will discuss a little history of the conjecture, construct our new groups V_T[N], and discuss how hyperbolic groups embed in the V_T[N] and how the V_T[N] embed in finitely presented simple groups.