Spring 2019

Abstract: I will describe what a relatively hyperbolic group is, and give a lot of examples where the boundary is planar. Furthermore, I will explore some of the interesting phenomena that can occur and explain the significance of cut points in the boundary. Lastly I will discuss restrictions on the peripheral groups when the boundary is planar and without cut points.This is joint work in progress with Chris Hruska

Abstract: Fiber bundles with fiber a surface arise in many areas including hyperbolic geometry, symplectic geometry, and algebraic geometry. Up to isomorphism, a surface bundle is completely determined by its monodromy representation, which is a homomorphism to a mapping class group. This allows one to use algebra to study the topology of surface bundles. Unfortunately, the monodromy representation is typically difficult to “compute” (e.g. determine its image). In this talk, I will discuss some recent work toward computing monodromy groups for holomorphic surface bundles, including certain examples of Atiyah and Kodaira. This can be applied to the problem of counting the number of ways that certain 4-manifolds fiber over a surface. This is joint work with Nick Salter.

Abstract: A group is said to have uniform exponential growth if the number of elements that can be spelled with words of bounded length is bounded below by a single exponential function over all generating sets. In 1981, Gromov asked whether all groups with exponential growing group in fact have uniform exponential growth. While this was shown not to be the case in general, it has been answered affirmatively for many natural classes of groups such as hyperbolic groups, linear groups, and the mapping class groups of a surface. In 2018, Kar-Sageev show that groups acting properly on 2-dimensional CAT(0) cube complexes by loxodromic isometries either have uniform exponential growth or are virtually abelian by explicitly exhibiting free semigroups whose generators have uniformly bounded word length whenever they exist. These free semigroups witness the uniform exponential growth of the group. I will explain how certain arrangements of hyperplane orbits can be used to build loxodromic isometries generating free semigroups and then describe how to use the convex hull of their axes and the Bowditch boundary to extend Kar and Sageev’s result to CAT(0) cube complexes with isolated flats.

This is joint work with Radhika Gupta and Kasia Jankiewicz.

Abstract:In a Gromov hyperbolic space, geodesics satisfies the so-called Morse property. This means that if a geodesic and a quasi-geodesic share endpoints, then their Hausdorff distance is uniformly bounded. Remarkably, this is an equivalent characterization of hyperbolic spaces, meaning that all consequences of hyperbolicity can be ascribed to this property.
Using this observation to understand hyperbolic-like behaviour in spaces which are not Gromov hyperbolic has been a very successful idea, which led to the definition of important geometric objects such as the Morse boundary and stable subgroups.
Another strong consequence of hyperbolicity is the fact that local quasi-geodesics are global quasi-geodesics. This allows detecting global properties on a local scale, which has far-reaching consequences.
The goal of this talk is twofold. Firstly, we will prove results that are known for hyperbolic groups in a class of spaces satisfying generalizations of the above properties. Secondly, we show that the set of such spaces is large and contains several examples of interest, i.e. CAT(0) spaces and hierarchically hyperbolic spaces.

Abstract: Corresponding to each automorphism of a free group, there is a cyclic semi-direct extension of the free group. Gersten in 1994 gave an example of such a free-by-cyclic group which is a “poison subgroup” for non-positive curvature—any group containing it cannot act geometrically on a CAT(0) space. His argument is both beautiful and geometric. Gersten’s proof provides context for our result; we show that in contrast with the general case, many free-by-cyclic groups associated to polynomially-growing palindromic automorphisms do admit a geometric action on a CAT(0) space.

Abstract: In this talk I will introduce a procedure to produce interesting examples of non-positively curved cube complexes. The construction we suggest takes as input two finite simplicial complexes and gives as output a finite cube complex whose local geometry can be easily described. This local information can then be used to obtain global information, e.g. about cohomogical dimension and hyperbolicity of the fundamental group of the cube complex. This is joint work with Rob Kropholler.

Abstract: A well-know conjecture is that all finitely presented groups have semistable fundamental group at . A class of groups whose members have not been shown to be semistable at is the class of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class naturally partitions into two non-empty subclasses; those that have “bounded” and “unbounded” depth. We show those of bounded depth have semistable fundamental group at . Ascending HNN extensions produced by Ol’shanskii-Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at .

Abstract: In this talk, I will outline the entropy technique, which is a well-established component of the theory of random walks on discrete groups developed in the late 70s. In a breakthrough result in the mid-90s, Kaimanovich applied it to the problem of identification of a non-trivial Poisson boundary (bounded harmonic functions) in geometric or combinatorial terms. A distinct advantage of this theory is that it reduces difficult qualitative questions about the asymptotic behavior of random walks to rather easy quantitative estimates. In spite of this, extensions of this theory to the case of random walks on non-discrete topological groups remained quite rudimentary in the last three decades.

I will show how to fill the gap and extend the entropy technique to random walks on locally compact groups. This is joint work with Giulio Tiozzo from the University of Toronto.

Abstract: Graph products of abelian groups simultaneously generalize and interpolate between right-angled Coxeter groups and right-angled Artin groups. In this talk we will discuss the structure of their automorphism groups by extending techniques developed in the RAAG setting. In particular, we will show that the automorphism group of a graph product of abelian groups satisfies the Tits’ alternative and is residually finite. We will also prove a dichotomy, the outer automorphism group of a graph product of abelian groups is either virtually nilpotent or contains a non-abelian free subgroup, and provide graphical criteria to distinguish between these two cases. This is joint work with Andrew Sale.