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

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.

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.

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.

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).

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.

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.

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)

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.

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.

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.