Artin Groups, CAT(0) geometry and related topics

Schedule July 5-8, 2021

Slides from talks are here

  Monday Tuesday   Wednesday   Thursday   Friday
Geartown welcome coffee Geartown social mixer 8:30-
Geartown social mixer 9:00-
Geartown social mixer 8:30-
Geartown social mixer
Karen Vogtmann Goulnara Arzhantseva 9:00-
Koji Fujiwara 9:30-
Bert Wiest 9:00-
Fanny Kassel
break     10:00-
Michah Sageev break   10:00-
Martin Bridson
*Carolyn Abbott Jingyin Huang break   10:45-
*Kim Ruane break  
Lunch     11:15-
*Giovanni Paolini Lunch   11:15-
*Jon McCammond
*Corey Bregman *Thomas Koberda Lunch   1:15-
*Matt Cordes    
Pallavi Dani Remi Coulon 2:00-
Geartown office hours 2:15-
Jing Tao    
coffee         coffee      
Dani Wise Mladen Bestvina     4:00-
Piotr Przytycki    
  Banquet (outdoor)   Rain date for banquet. 5:00-
Zoom game night    
* indicates in-person talk
Virtual events. Links will be sent by email to registered participants.


Carolyn Abbott

Title: Generalizing quasiconvex subgroups of hyperbolic groups

Abstract: Hyperbolic groups are a well-studied class of groups which satisfy many nice algebraic and geometric properties. Their quasiconvex subgroups are those that sit ``nicely'' inside the group and so inherit many of the same properties. The study of quasiconvex subgroups of hyperbolic groups has been shown to be a powerful tool in geometric group theory. Unfortunately, the notion of a quasiconvex subgroup is not well-defined for groups that are not hyperbolic. In this talk, I will present a generalization of a quasiconvex subgroup to the class of acylindrically hyperbolic groups. This is a large class of groups which includes mapping class groups, right-angled Artin and Coxeter groups, fundamental groups of many 3? manifolds, and Out(F_n), and many others. I will give some examples of these subgroups and show that they satisfy many of the same algebraic and geometric properties as quasiconvex subgroups of hyperbolic groups. This is joint work with Jason Manning..

Goulnara Arzhantseva

Title: Approximations of infinite groups

Abstract: We discuss various (still open) questions on approximations of finitely generated groups, focusing on finite-dimensional approximations such as residual finiteness and soficity. We survey our results on the existence and stability of metric approximations. We suggest a few conjectures, e.g. on Gromov hyperbolic groups and their infinite monster limits.

Based on joint works with Liviu Paunescu (Bucharest).

Mladen Bestvina

Title: The Farrell-Jones conjecture for hyperbolic-by-cyclic groups

Abstract: Most of the talk will be about the Farrell-Jones conjecture from the point of view of an outsider. I'll try to explain what the conjecture is about, why one wants to know it, and how to prove it in some cases. The motivation for the talk is my recent work with Fujiwara and Wigglesworth where we prove this conjecture for (virtually torsion-free hyperbolic)-by-cyclic groups. If there is time I will outline the proof of this result.

Corey Bregman

Title: Outer Space for RAAGs

Abstract: Right-angled Artin groups (RAAGs) range from free groups to free abelian groups, and thus their outer automorphism groups share properties with both Out(F_n) and GL(n,Z). Over the past decade, automorphisms of RAAGs have been well-studied, though largely from an algebraic point of view. To allow for a more geometric approach, in this talk we construct an outer space for RAAGs generalizing Culler-Vogtmann outer space for Out(F_n) and the classical symmetric space of positive definite inner products for GL(n,Z). To do this, we will describe a family of locally CAT(0) metric spaces built from parallelotopes, whose fundamental group is a given RAAG. We then show that the space of all such (marked) metrics is finite-dimensional, contractible, and admits a proper action of the outer automorphism group. This is joint work with Ruth Charney and Karen Vogtmann.

Martin Bridson

Title: Sidki doubles and subdirect products of groups

Abstract: The Sidki double X(G) of a group G is obtained from the free product G*G by requiring each element g in the first copy of G to commute with the same g in the second copy of G. I shall explain why X(G) is finitely presented if G is finitely presented, and why this construction is connected to subdirect products of groups. We shall then explore the potential of this new source of finitely presented groups. This is joint work with Dessislava Kochloukova (Campinas).

Remi Coulon

Title: First person exploration of Thurston's geometries

Abstract: Thurston's conjecture (proved by Perelman) states that any reasonable three-dimensional space can be decomposed into elementary "building blocks" each of which is modeled onto a specific geometry. There are eight such models. Some of them are familiar, like the Euclidean space or the hyperbolic space, others are wilder like Nil or Sol. To gain more insight on this topic, we develop a web app that simulates in real-time what an inhabitant would see in each of these geometries.

In this talk we will explain how this software was built and exploit it to illustrate some fun features of the Thurston geometries.

This is a joint work with Elisabetta A. Matsumoto, Henry Segerman, and Steve Trettel

Matt Cordes

Title: Geometric approximate group theory

Abstract: An approximate group is a subset of a group that is "almost closed" under multiplication. Finite approximate subgroups play a major role in additive combinatorics. In 2012, Breuillard, Green and Tao established a structure theorem concerning finite approximate subgroups and used this theory to reprove Gromov's polynomial growth theorem. Infinite approximate groups were studied implicitly long before the formal definition. Meyer, in 1972, developed a theory of mathematical quasi-crystals which can be seen as approximate subgroups of R^n that are Delone. Recently, Bjorklund and Hartnick have begun a program investigating infinite approximate lattices in locally compact second countable groups using geometric and measurable structures. In the talk I will introduce infinite approximate groups and their geometric aspects. This is joint work with Hartnick and Toni.

Pallavi Dani

Title: Subgroups of right-angled Coxeter groups via Stallings-like techniques

Abstract: Stallings folds have been extremely influential in the study of subgroups of free groups. I will talk about joint work with Ivan Levcovitz, in which we develop an analogue for the setting of right-angled Coxeter groups. I will describe several applications, including a recent new construction of non-quasiconvex subgroups of hyperbolic groups.

Koji Fujiwara

Title: The rates of growth in a hyperbolic group

Abstract: I discuss the set of rates of growth of a hyperbolic group with respect to all its finite generating sets. It turns out that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. This is a joint work with Sela.

Jingyin Huang

Title: Rigidity of some large type Artin groups

Abstract: For a large class of large type Artin groups, Crisp (2005) introduced an analogue of ``curve graphs'' for them, and proved that automorphism of these curve graphs are induced by the automorphisms of the underlying Artin groups - which can be viewed as an analogue of the well known Ivanov's theorem for mapping class groups. As Ivanov's theorem is closely related to strong quasi-isometric rigidity and measure equivalence rigidity of mapping class groups, we study how would Crisp's theorem lead to strong quasi-isometric rigidity and measure equivalence rigidity of large type Artin groups. In this talk, I will survey some rigidity results in this direction, discuss some geometric ingredients in the proof, as well as some open questions. This talk is based on joint work with Damian Osajda, and independently, with Camille Horbez.

Fanny Kassel

Title: Convex cocompactness for Coxeter groups

Abstract: Representations of Coxeter groups as linear reflection groups, a la Vinberg, yield interesting examples of convex projective orbifolds. In joint work with J. Danciger, F. Gueritaud, G.-S. Lee, and L. Marquis, we determine in which cases (namely, for which Coxeter groups and which representations) these orbifolds are convex cocompact. This includes all cases where the Coxeter group is word hyperbolic and the representation is a so-called Anosov representation (a notion playing an important role in higher Teichmuller theory), but also other cases where the Coxeter group is not hyperbolic.

Thomas Koberda

Title: Combinatorics of graphs via right-angled Artin groupsle

Abstract: I will discuss the problem of extracting combinatorial information about a graph from the group theory of the associated right-angled Artin group. I will concentrate on the problem of deciding whether or not a graph admits a Hamiltonian cycle. This talk represents joint work with Ramón Flores and Delaram Kahrobaei.

Jon McCammond

Title: Dual Braids and Braid Arrangement

Abstract: The braid groups have two well known Garside presentations. The elegant minimal standard presentation is closely related to the Salvetti complex, a cell complex derived from the complement of the complexification of the real braid arrangement. The dual presentation, introduced by Birman, Ko and Lee, leads to a second Garside structure and a second classifying space, but it has been less clear how the dual braid complex is related to the (quotient of the) complexified hyperplane complement, other than abstractly knowing that they are homotopy equivalent. In this talk, I will discuss recent progress on this issue. Following a suggestion by Daan Krammer, Michael Dougherty and I have been able to embed the dual braid complex into the complement of the complex braid arrangement. This leads in turn to a whole host of interesting complexes, combinatorics, and connections to other parts of the field. This is joint work with Michael Dougherty.

Giovanni Paolini

Title: The K(π,1) conjecture for affine Artin groups

Abstract: I will outline the recent proof of the K(π,1) conjecture for affine Artin groups, which is a joint work with Mario Salvetti. The proof makes use of the dual presentations of (affine) Coxeter and Artin groups while mixing combinatorial, topological, and geometric arguments.

Piotr Przytycki

Title: Tits Alternative in dimension 2

Abstract: A group G satisfies the Tits Alternative if each of its finitely generated subgroups contains a non-abelian free group or is virtually solvable. I will sketch a proof of a theorem saying that if G acts geometrically on a simply connected nonpositively curved complex built of equilateral triangles, then it satisfies the Tits Alternative. This is joint work with Damian Osajda.

Kim Ruane

Title: Boundary Rigidity for Lattices in a Product of Trees

Abstract: This is joint work with K. Jankiewicz, Annette Karrer, and Bakul Sathaye. We are interested in showing that if G is a lattice in a product of locally finite trees acting vertex-transitively and freely, then G is boundary rigid - i.e. if such a G acts geometrically on another CAT(0) space X, then the visual boundary of X is a join of two Cantor sets. We first consider the case where X is CAT(0) cubical because the Rank Rigidity Theorem holds in that setting allowing for a streamlined proof of the result. In the general case, there are interesting difficulties getting around this that I will discuss in the talk.

Michah Sageev

Title: Actions of right angled Coxeter groups on CAT(0) cube complexes

Abstract: As right angled Coxeter groups and CAT(0) cube complexes are two of Ruth?s favorite topics, we will talk about actions of right angled Coxeter groups on CAT(0) cube complexes.

Jing Tao

Title: Big mapping class groups with abelianizations big and small

Abstract: Big mapping class groups, or mapping class groups of surfaces of infinite type, form a rich class of non-locally compact Polish (separable, completely metrizable) groups. The behavior of these groups can be quite wild. In this talk, I will showcase a large class of examples that are perfect and another large class of examples whose abelianizations can map onto an infinitely-generated abelian group. This is joint with Justin Malestein.

Karen Vogtmann

Title: Graph counting and the integral Euler characteristic of Out(F_n)

Abstract: Methods for counting graphs inspired by quantum field theory have given new algorithms for and new qualitative information about the rational Euler characteristic of Out(F_n). I will describe this and also tell about recent progress on the subtler question of determining the integral Euler characteristic. This is joint work with M. Borinsky.

Bert Wiest

Title: Morse elements in Garside groups are strongly contracting

Abstract: We prove that in the Cayley graph of any braid group modulo its center B_n/Z(B_n), equipped with Garside's generating set, the axes of all pseudo-Anosov braids are strongly contracting. More generally, we consider a Garside group G of finite type with cyclic center. We prove that in the Cayley graph of SG/Z(G), equipped with the Garside generators, the axis of any Morse element is strongly contracting. As a consequence, we prove that Morse elements act loxodromically on the additional length complex of G. This is joint work with Matthieu Calvez.

Dani Wise

Title: A graph coloring problem and the virtual fibering of right-angled Coxeter groups.

Abstract: I will first describe a simple graph coloring problem and survey some examples of graphs for which the coloring problem has or has no solution. Through Bestvina-Brady Morse theory, this coloring problem relates to the virtual algebraic fibering for right-angled Coxeter groups. For instance we virtually algebraically fiber some hyperbolic 4-manifolds. Finally, I will discuss the recent work of Italiano-Martelli-Migliorini who applied a variation of these ideas to obtain virtual algebraic fibering of some 5-manifolds, but with finite-type fiber. This is joint work with Kasia Jankiewicz and Sergey Norin.