Tag - Artin groups

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Vladimir Vankov: Bestvina-Brady groups and generalizations

5th June, 2023

Right-angled Artin groups are perhaps the most ubiquitous manifestations of polyhedral products in geometric group theory and low-dimensional topology. The theory of their subgroups has been of great importance in the last couple of decades. This is especially true with regards to what are known as 'finiteness properties' - meaningful criteria for measuring ways in which infinite groups may behave like finite ones - as well as the theory of three-dimensional manifolds. We will visit some celebrated theorems and, if time allows, discuss problems arising from deck transformations of branched covering maps.

Thomas Koberda: Hamiltonicity of graphs via right-angled Artin groups

1st April, 2022

I will discuss the dictionary between the algebraic structure of a right-angled Artin group and the combinatorics of the defining graph. I will then use the cohomology of a right-angled Artin group to provide a characterization of Hamiltonicity of the underlying graph. Along the way, I will describe some linear algebraic facts which appear to have been previously unknown.

Jingyin Huang: The Helly geometry of some Garside and Artin groups

15th November, 2021

Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required.

Alessandro Sisto: A simple hierarchical hyperbolicity criterion and extra-large Artin groups

5th November, 2021

A hierarchically hyperbolic structure is some kind of coordinate system on a given metric spaces where the coordinates take values in hyperbolic spaces, and it gives a good understanding of the coarse geometry of the space. I will give a brief introduction to this notion and its consequences, discuss a simple criterion to show that a space or group is hierarchically hyperbolic, and illustrate an application of this criterion to the case of extra-large type Artin groups.

Rose Morris-Wright: Artin groups: algebraic and geometric techniques

28th June, 2021

Artin groups are a broad class of groups whose presentations all follow a particular pattern. They are generalizations of braid groups and are closely related to Coxeter groups. Artin groups provide examples of groups with many interesting properties but there is very little that is known about ALL Artin groups.

In the first lecture, we’ll define Artin groups, talk about different types of Artin groups, and give a summary of known results and open questions. In the second lecture, we’ll focus on algebraic techniques for studying Artin groups, including the Garside structure, parabolic subgroups, and, if time permits, the Artin monoid. In the third lecture, we’ll discuss geometric techniques for studying Artin groups, including the Deligne complex and newer complexes such as the Clique-cube complex and the systolic Artin complex.

Mark Hagen: Hierarchical hyperbolicity from actions on simplicial complexes

5th June, 2020

The notion of a "hierarchically hyperbolic space/group" grows out of geometric similarities between CAT(0) cubical groups and mapping class groups. Hierarchical hyperbolicity is a "coarse nonpositive curvature" property that is more restrictive than acylindrical hyperbolicity but general enough to include many of the usual suspects in geometric group theory. The class of hierarchically hyperbolic groups is also closed under various procedures for constructing new groups from old, and the theory can be used, for example, to bound the asymptotic dimension and to study quasi-isometric rigidity for various groups. One disadvantage of the theory is that the definition - which is coarse-geometric and just an abstraction of properties of mapping class groups and cube complexes - is complicated. We therefore present a comparatively simple sufficient condition for a group to be hierarchically hyperbolic, in terms of an action on a hyperbolic simplicial complex. I will discuss some applications of this criterion to mapping class groups and (non-right-angled) Artin groups.

Federico Berlai: From hyperbolicity to hierarchical hyperbolicity

5th June, 2020

Hierarchically hyperbolic groups (HHGs) and spaces are recently introduced generalizations of (Gromov-) hyperbolic groups and spaces. Other examples of HHGs include mapping class groups, right-angled Artin/Coxeter groups, and many groups acting properly and cocompactly on CAT(0) cube complexes. After a substantial introduction and motivation, I will present a combination theorem for hierarchically hyperbolic groups. As a corollary, any graph product of finitely many HHGs is itself a HHG.

Zoran Šunić: Deciding if a right-angled Artin group is free-by-free is NP-complete

19th March, 2020

We show that deciding if a right-angled Artin group is free-by-free is an NP-complete problem. The work is based on an earlier result by Susan Hermiller and the speaker stating that the right-angled Artin group AΓ defined by the graph Γ is free-by-free if and only if Γ is 2-breakable (a graph Γ is 2-breakable if there exists an independent set D of vertices in Γ such that every cycle in Γ contains as least two vertices from D). We reduce the 3SAT Problem to the problem of deciding if a given graph is 2-breakable (in fact, k-breakable, for any fixed k ≥ 1). Once it is shown that the problem is NP-complete, it is not difficult to show that it stays NP-complete even if we restrict it to right-angled Artin groups defined by planar graphs. Note that the more special problem of deciding if a right-angled Artin group is free-by-infinite-cyclic has a very simple answer. Namely, it follows easily from known results that the following three statements are equivalent. (1) AΓ is free-by-infinite-cyclic. (2) Γ is a forest. (3) AΓ embeds in the right angled group defined by the path of length 3. (Joint work with David Carroll and Benjamin Francisco.)