Group actions on hyperbolic spaces

In this talk we survey the properties of groups acting on hyperbolic metric spaces from several perspectives: algebraic, geometric, algorithmic, and probabilistic. We define basic notions such as hyperbolic metric spaces, Cayley graphs, word metrics, and quasi-isometries. We then survey properties of hyperbolic groups. Finally, we give an introduction to mapping class groups and the curve graph and survey some results on mapping class groups that are proven using curve graph techniques. The techniques applicable to mapping class groups are applicable to many other families of groups, and we mention some of these families at the end.

Stretch factor of free group automorphisms and free-by-cyclic groups 

Given a graph map from a graph to itself, we can associate two numbers to it: geometric stretch factor and homological stretch factor. I will define a notion of orientability for graph maps and use it to characterise when the two numbers are equal. The notion of orientability can be upgraded for (fully irreducible) automorphisms of free groups as well, motivated by stretch factors of pseudo-Anosov surface homeomorphisms. Time permitting, we'll then talk about free-by-cyclic groups and see how the notion of orientability interacts with the cohomology classes in the BNS invariant of the free-by-cyclic group. This is joint work with Spencer Dowdall and Samuel Taylor.

Paper: Orientable maps and polynomial invariants of free-by-cyclic groups, with Spencer Dowdall and Samuel Taylor 

Morse boundaries of graphs of groups with examples of right-angled Artin and Coxeter groups

Associated to a finitely generated group is a topological space called Morse boundary introduced by Chareny--Sultan and Cordes. This boundary indicates how similar the group is to a Gromov-hyperbolic group. We will study the question of how the Morse boundary of a graph of groups is related to the Morse boundary of its vertex groups. To do so we will consider examples of right-angled Artin- and Coxeter groups.

In the first talk, we will study examples of Morse boundaries of right-angled Coxeter and Artin groups. These groups are defined by graphs. We will investigate how certain decompositions of the defining graphs are related to the Morse boundaries of the corresponding groups. In particular, we will learn when the Morse boundary of a right-angled Coxeter group with totally disconnected Morse boundary remains totally disconnected if we glue certain graphs on its defining graph.

In the second part of the talk, we will see a generalization of the first part to Morse boundaries of graphs of groups. Thereby, the edge groups don't contribute to the Morse boundary of the ambient group. I will finish the second talk with examples showing what can happen if we lose this restriction on the edge groups.

The first talk is based on my dissertation. The second talk is joint work with Elia Fioravanti and joint work with Marius Graeber, Nir Lazarovich, and Emily Stark.

Papers: 

1. Right-angled Coxeter groups with totally disconnected Morse boundaries 

2. Connected components of Morse boundaries of graphs of groups, with Elia Fioravanti 

3. Surprising circles in Morse boundaries of right-angled Coxeter groups, with Marius Graeber, Nir Lazarovich, and Emily Stark

Pattern Preserving Quasi-isometries in Lamplighter Groups

Lamplighter groups have a Cayley graph which can be constructed as a subset of a product of two trees. We define pattern preserving quasi-isometries to be quasi-isometries of the Cayley graph that act bijectively on set of cosets of certain subgroup. A similar construction can be made for the SOL and Baumslag Solitar group, and in these cases results by Schwartz(1996) and Taback(1998) show that pattern preserving quasi-isometries have strong rigidity properties. In this talk, I will compare the lamplighter case to these two known cases, and explain how pattern preserving quasi-isometries in the lamplighter group are in some ways similar to the SOL and Baumslag Solitar cases, and is some ways much less rigid. (Joint work with Tullia Dymarz, Natasa Macura, and Beibei Liu.)

Hyperbolic models for CAT(0) spaces

The goal of the talk is to introduce analogues of curve graphs and cubical hyperplanes for the class of CAT(0) spaces. As applications we will sketch a dichotomy of rank-rigidity flavour, characterizations of rank-one elements and, if time allows, an analogue of Ivanov’s theorem. This is joint work with H. Petyt and A. Zalloum.

Paper: Hyperbolic models for CAT(0) spaces, with H. Petyt and A. Zalloum.

Outer automorphisms of property (T) groups

The combination of Kazhdan's property (T) and negative curvature typically limits the amount of outer automorphisms. Indeed, it is a result of Paulin that every property (T) hyperbolic group has a finite outer automorphism group. Belegradek and Szczepanski extends Paulin's result to property (T) relatively hyperbolic groups. We prove that for every countable group Q there is an acylindrically hyperbolic group G such that Out(G)=Q. Therefore the combination of property (T) and acylindrical hyperbolicity is much more flexible in terms of outer automorphisms. This is a joint work with Ionut Chifan, Adrian Ioana and Denis Osin.

Commensurating HNN-extensions and biautomaticity

Biautomatic groups, introduced in the early 1990s, arose as groups explaining formal language-theoretic aspects of geodesics in word-hyperbolic groups. Many classes of non-positively curved finitely generated groups, such as hyperbolic, virtually abelian, cocompactly cubulated and Coxeter groups, are known to be biautomatic. For a long time, it was believed that all CAT(0) groups might possibly be biautomatic, but this has recently been shown not to be the case. It has also been open if all hierarchically hyperbolic groups (HHGs) -- a large class containing hyperbolic, most cocompactly cubulated, and mapping class groups -- are biautomatic.

In this talk I will study biautomaticity in the context of two families of groups acting on locally finite trees. For the first of these families -- commensurating HNN-extensions of free abelian groups, also known as Leary--Minasyan groups -- I will explain why they give examples not only of non-biautomatic CAT(0) groups, but also of CAT(0) groups that cannot appear as subgroups of biautomatic groups. Based on joint work with Sam Hughes, I will also explain why the second family -- commensurating HNN-extensions of surface groups -- contains an example of a non-biautomatic HHG that satisfies some other curious geometric properties, such as being injective but not Helly. The proofs of non-biautomaticity rely on studying the asymptotic properties of translation lengths for biautomatic structures on abelian and surface groups.

Papers: 

1. Leary-Minasyan subgroups of biautomatic groups. 

2. Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity, with S. Hughes.