# McGill Geometric Group Theory Seminar

McGill GGT research group

### The seminar meets each Wednesday at 3 PM in 920 Burnside Hall

Seminar organizers: Nima Hoda, Kasia Jankiewicz, Piotr Przytycki, Daniel Wise

## Upcoming talks:

#### October 25, 2017: Eduard Einstein (Cornell University), Hierarchies of Non-Positively Curved Cube Complexes.

A non-positively curved (NPC) cube complex is a combinatorial complex constructed by gluing Euclidean cubes along faces in a way that satisfies a combinatorial local non-positive curvature condition. A hierarchy is an inductive method of decomposing the fundamental group of a cube complex. Cube complexes and hierarchies of cube complexes have been studied extensively by Wise and feature prominently in Agol's proof of the Virtual Haken Conjecture for hyperbolic 3-manifolds. In this talk, I will give an overview of the geometry of cube complexes, explain how to construct a hierarchy for a NPC cube complex, and discuss applications of cube complex hierarchies to hyperbolic and relatively hyperbolic groups.

#### November 8, 2017: Matthew Zaremsky (University at Albany - SUNY), Finiteness properties in the extended family of Thompson's groups.

I will discuss a variety of groups in the extended family of Thompson's groups, some natural CAT(0) cube complexes on which they act, and the Morse theoretic tools that lead to understanding their finiteness properties. This will span various projects of mine, often joint with combinations of Kai-Uwe Bux, Martin Fluch, Marco Marschler, Lucas Sabalka, Rachel Skipper, and Stefan Witzel. Prior knowledge of Thompson's groups will not be assumed.

#### November 15, 2017: Daniel Wise (McGill University), No growth-gaps for special cube complexes I.

This will consist of two lectures, the first by Wise and the second by Li on their joint work.

Let be G the fundamental group of a compact special cube complex. We show that there is a sequence of infinite index quasiconvex subgroups whose growth-rates converge to the growth-rate of G.

In the first talk, we will discuss growth-rates, prove the result in the case when G is a free group (work of Dahmani-Futer-Wise), sketch a proof of the general case. In the second talk, we will give details of the proof, including the automatic structure that it relies upon and the method for using automata to estimate growth-rates. We will also survey a collection of problems that are motivated by this work.

## Past talks:

#### September 6, 2017: Stéphane Lamy (Université Paul Sabatier), Signature morphisms on the Cremona group.

The Cremona group is the group of birational symmetries of the plane. This is a rather huge group, which is neither finitely generated nor finite dimensional. The abelianization of the Cremona group was recently studied: over the base field $\mathbb{C}$, the abelianization is trivial, but over $\mathbb{R}$, the abelianization is an uncountable direct sum of $\mathbb{Z}/2$. I will discuss similar results over more general fields, such as number fields or finite fields. The key is to use a natural action on a square complex, to exhibit various trees as quotients or subcomplexes, and then to apply Bass-Serre theory (joint work with Susanna Zimmermann).

#### September 13, 2017: Piotr Przytycki (McGill University), A step towards Twist Conjecture.

I will explain Muehlherr's Twist Conjecture describing the hypothetical solution to the isomorphism problem in Coxeter Groups. I will mention recent progress in verifying it which is joint work with Jingyin Huang.

#### September 20, 2017: Nima Hoda (McGill University), Geocyclic Groups.

A graph is geocyclic if there is an upper bound on the lengths of its isometrically embedded cycles. The 1-skeleta of many combinatorially nonpositively curved spaces are geocyclic. I will present these examples and explain why a finitely generated group admitting a geocyclic Cayley graph is finitely presented and has solvable word problem. Time permitting, I will describe a strengthened form of geocyclicity and use it to prove that every Cayley graph of $\mathbf{Z}^2$ is geocyclic. This is joint work with Daniel Wise.

#### September 27, 2017: Sang-hyun Kim (Seoul National University), Free products in $\operatorname{Diff}(S^1)$.

We prove that if $G$ is a finitely generated non-virtually-abelian group, then $(G \times \mathbb{Z}) \ast \mathbb{Z}$ does not embed into $\operatorname{Diff}^2(S^1)$. In particular, the class of subgroups of $\operatorname{Diff}^r(S^1)$ is not closed under taking free products for each $r \ge 2$. We complete the classification of RAAGs embeddable in $\operatorname{Diff}^r(S^1)$ for each $r$, answering a question in a paper of M. Kapovich. (Joint work with Thomas Koberda)

#### October 4, 2017: Daniel Wise (McGill University), Characterizing Residual Finiteness of Tubular Groups.

A tubular group is a group that splits as a graph of groups where all edge groups are $\mathbf{Z}$ and all vertex groups are $\mathbf{Z}\times\mathbf{Z}$. There are many interesting examples of tubular groups exhibiting subtle and pathological behaviour. In my talk I will characterize the f.g. tubular groups that are residually finite. This is (ongoing) joint work with Daniel Woodhouse.

#### October 11, 2017: Bogdan Nica (McGill University), Proper isometric actions of hyperbolic groups on Lp-spaces, revisited.

There are several known constructions of actions as in the title (Yu, Nica, Bourdon, Alvarez - Lafforgue). I will discuss a new construction which, apart from a technical point, is particularly simple.

#### October 18, 2017: Jonah Gaster (McGill University), Sageev's cube complex dual to a collection of curves and lengths of curves on hyperbolic surfaces.

Sageev gave a very general construction of a CAT(0) cube complex dual to a `space with walls', and this construction has proved extraordinarily useful in recent celebrated work of Agol, Wise, and others. In one of the simplest nontrivial settings, this construction produces a non-positively curved cube complex dual to a finite collection of non-homotopic essential closed curves on a surface. I will show how this cube complex can be used to analyze the length function associated to a system of curves on the moduli space of hyperbolic structures on a surface of genus g, adding context to previous work of Basmajian.

Seminar schedule archive