On the dual approach and non-crossing partitions

María Cumplido (Sevilla): Parabolic subgroups in Artin groups
Given an Artin group with its standard presentation, a standard parabolic subgroup is the subgroup generated by a subset of generators. A parabolic subgroup is any conjugate of a standard parabolic subgroup. During this mini-course, we will try to explain why these subgroups are crucial in the study of the properties of Artin groups and the complexes on which they act, as well as the main techniques that have enabled an explosion of new results on these objects in the last decade.

Thomas Gobet (Clermont-Ferrand): Garside groups and Burau representations
Since the seminal work of Krammer, several actions of braid groups have been shown to be faithful using Garside-theoretic tools. This includes many linear actions, and also actions on categories. Given a Garside group G and a representation V of G, to show faithfulness it suffices to show that the restriction of the representation to the Garside monoid M is faithful. To this end one often tries to reconstruct the Garside normal form of an element of the monoid from its matrix in a suitable basis of the representation.
In the first lecture we will explain how faithfulness of the reduced Burau representation of a dihedral Artin group can be shown using this philosophy, mostly following a proof of Lehrer and Xi (building on ideas of Krammer, and using techniques also developed independently by other authors).
In the second lecture we will construct a ”Burau representation” for another important family of Garside groups: torus knot groups. We will explain how these representations can be shown to be faithful using the aforementioned philosophy. As a byproduct this gives an explicit construction of the Burau representation for some complex braid groups of rank two, and for toric reflection groups.

Eddy Godelle (Caen): Cactus groups associated with parabolic subgroups
The object of the two lectures is to introduce cactus groups and to explain how they are related to Coxeter groups and Artin groups.
In the first talk, we will see, on the one hand, why they are closed to braid groups from a topological point of view, and on the other hand, why they are naturally related to Coxeter groups and their family of parabolic subgroups from a combinatorial point of view. During the second talk we will explain how one can solve the word problem for these groups, introducing the notion of a trickle presentation. We will then see how to associate a Cactus group with Dual presentations of Artin groups.

Jon McCammond (Santa Barbara): The geometric combinatorics of polynomials and braids
The space of monic centered degree d polynomials with distinct roots is one of the original classifying spaces for the braid groups (and the model for the construction of general spaces whose fundamental groups are general Artin groups). In this minicourse I will describe how the geometry and combinatorics of polynomials can be used to define a finite piecewise Euclidean cell complex that (1) is related to the dual Garside structure for braid groups and (2) can be viewed as a compactification of the original polynomially-defined classifying space for the braid groups. This is joint work with Michael Dougherty.

Luis Paris (Dijon): Automorphism groups of Artin groups of spherical type
Recall that an Artin group is of spherical type if its associated Coxeter group is finite. The aim of this mini-course is to give an overview on what is known on the automorphism groups of Artin groups of spherical type. Some ideas and techniques in the proofs as well as some open questions will be given.

Federica Gavazzi (Dijon): Spaces related to virtual Artin groups
Virtual Artin groups, denoted VA[Γ], were introduced by Bellingeri, Paris, and Thiel to extend the ”virtual” concept from braids to all Artin groups. As in the braid case, there are two homomorphisms to the Coxeter group, which define the pure and Kure virtual Artin groups. This talk will delve into the topology of these groups by constructing spaces associated with them. We will sketch the construction of a CW-complex Ω(Γ), whose fundamental group is isomorphic to the pure virtual Artin group PVA[Γ], generalizing the existing so-called BEER complex from virtual braids to all Coxeter graphs. We claim that Ω(Γ) is aspherical when Γ is of spherical type, thereby making it a classifying space for PVA[Γ] in this context. To achieve this result, we link Ω(Γ) with the Salvetti complex of a related Coxeter graph Γ', connecting its asphericity to the K(π, 1)-conjecture for Γ'. A new class of “almost parabolic” reflection subgroups of the Coxeter group plays a crucial role in this construction.

Igor Haladjian (Tours): Braid groups of virtual complex reflection groups of rank two and associated Garside Structures
After introducing virtual complex reflection groups of rank two, I will define their braid groups and present some results generalizing results about braid groups of irreducible complex reflection groups of rank two. Moreover, I will present a Garside structure for these braid groups which generalize those of rank two complex reflection groups, as well as new Garside structures for the braid groups associated to some rank two complex reflection groups.

Georges Neaime (Bielefeld): Classical and Elliptic Theories for Reflection and Artin Groups
We briefly describe the theory of elliptic reflection groups and their Artin groups and compare it with the classical theory. We will discuss some recent developments and applications of the elliptic theory, as well as challenging open problems.

Sarah Rees (Newcastle): Geodesics in Artin groups; application to the Deligne complex
Let G = ⟨X | R⟩ be an Artin group in its standard presentation, where each element of R has the form x_ix_jx_i … = x_jx_ix_j …, relating two alternating products of the same length m_ij , for distinct x_i, x_j ∈ X, m_ij ∈.
Work of mine with Derek Holt 12 years ago derived a system of rewrite rules involving only 2-generator strings, valid whenever that G had large type (all m_ij ≥ 3), or at least sufficiently large type. It could reduce any word over X to a geodesic representative in G in quadratic time, hence solving the word problem. As a result, we understood the structure of geodesics in many (but not all) 3-generated Artin groups; recent work of Blasco, Cumplido and Morris-Wright (which adapted our methods) then dealt with all the remaining 3-generated groups apart from those with parameters (2, 3, k), k > 5.
It turned out that this knowledge was useful to Boyd, Charney and Morris-Wright in their study of Deligne complexes, in which they exploit the relationship between the Deligne complex for an Artin group and the complex associated to its related monoid.
Considering that application, Derek Holt and I worked to extend the results of Blasco, Cumplido and Morris-Wright to cover those 3-generated Artin groups with parameters (2, 3, k) that were not covered by the existing results. More recently, in joint work with Maria Cumplido, we have extended the results of Blasco, Cumplido and Morris-Wright further, and have now found an effective rewrite system that solves the word problem in any Artin group whose diagram contains no A₃ or B₃ subdiagrams.
I shall discuss all of the above. Background and motivation will be provided, and technical terms explained.

Workshop Room: V2-205

	Monday	Tuesday	Wednesday	Thursday	Friday
09:00 – 10:00	Paris I	Godelle II	McCammond I	Cumplido I	Gobet II
10:00 – 10:45	Coffee	Coffee	Coffee	Coffee	Coffee
10:45 – 11:45	Godelle I	Paris II	McCammond II	Gobet I	Cumplido II
12:00 – 15:00	Lunch Free discussions
15:00 – 16:00	Common discussions	Common discussions	Free afternoon: Hike and dinner	Common discussions	Common discussions
16:00 – 17:00	Gavazzi	Neaime		Haladjian	Rees
17:30 –				Reception

Participants

Jad Abou-Yassin (Tours)
Barbara Baumeister (Bielefeld)
Charly Chwabe (Paderborn)
Maria Cumplido (Sevilla)
Federica Gavazzi (Dijon)
Thomas Gobet (Clermont-Ferrand)
Eddy Godelle (Caen)
Igor Haladjian (Tours)
Jon McCammond Santa (Barbara)
Georges Neaime (Bielefeld)
Marcel Palmer (Bielefeld)
Luis Paris (Dijon)
Sarah Rees (Newcastle)
Patrick Wegener (Hannover)

Organisers

Barbara Baumeister
Thomas Gobet
Georges Neaime