Negami Haru (Chiba University)

Constructing representation of the Artin groups of type A_n and KZ-type equation

In this talk, we introduce a correspondence between algebraic and analytic approaches to constructing representations of the braid group, the Artin group of type $A_n$. Starting from the left action of the braid group $B_n$ on the free group $F_n$, known as the Artin representation, we consider the semidirect product $F_n \rtimes B_n$. The Long-Moody construction provides a method to obtain linear representations of $B_n$ from representations of $F_n \rtimes B_n$. Extending this, the Katz-Long-Moody construction generates an infinite sequence of representations of $F_n \rtimes B_n$. Moreover, a natural transformation exists between the Katz-Long-Moody construction and multiplicative middle convolution associated with the Knizhnik-Zamolodchikov (KZ)-type equation.

Furthermore, we discuss a generalization introduced by Wada, who extended the Artin representation aiming to construct group invariants of knot. This generalization, called the Wada representation, has been classified by Ito. In this study, we further generalize the Katz-Long-Moody construction using the Wada representation and explore analytic correspondences and the unitarity of the representation.

Kuno Erika (Osaka University)

Right-angled Artin subgroups of mapping class groups for nonorientable surfaces and curve graphs

Koberda proved that for closed orientable surfaces $S=S_{g, n}$ of genus $g$ with $n$ marked points ($2-g-n<0$) if $\Gamma$ is a finite full subgraph of the curve graph $\mathcal{C}(S)$ of $S$ then the right-angle Artin group $A(\Gamma)$ of $\Gamma$ is a subgroup of the mapping class group ${\rm Mod}(S)$ of $S$. Let $N=N_{g,n}$ be a closed nonorientable surface of genus $g$ with $n$ marked points. We consider the full subgraph $\mathcal{C}_{\mathrm{two}}(N)$ of the curve graph $\mathcal{C}(N)$ which consists of the isotopy classes of the two-sided essential simple closed curves, and we call it the two-sided curve graph of $N$. We generalize the result of Koberda to the two-sided curve graph of $N$, that is, we prove that if $\Gamma$ is a finite full subgraph of

$\mathcal{C}_{\mathrm{two}}(N)$, then the right-angle Artin group $A(\Gamma)$ of $\Gamma$ is a sugroup of the mapping class group ${\rm Mod}(N)$ of $N$. This is a joint work with Takuya Katayama.

Maria Cumplido (University of Seville)

Rewritings to solve the word problem in big families of Artin groups

Artin groups are defined by a presentation of the form: 

A(S) = \langle S \mid m_{s,t}(s,t) = m_{s,t}(t,s) \text{ for all } s,t \in S \rangle,  

where $ m_{s,t}(s,t) $ represents a word of length $ m_{s,t} $ alternating between $s$ and $t$. Despite their easy presentation, many fundamental algorithmic problems in Artin groups remain open, including the word and conjugacy problems. Surprisingly, the techniques used to study these problems vary significantly depending on the family of Artin groups considered and finding homogenous tools is one of the big questions in the field. We will explain a method of rewriting words that allows us to obtain geodesic representatives for elements in 3-free Artin groups, namely those that do not have a relation of length 3. As a direct consequence, we solve the word problem in this family. Our approach is purely combinatorial and, therefore, particularly easy to implement. Our techniques are based on detecting a certain type of words, which we call critical, and defining from them transformation rules that we call $\tau$-moves. In particular, we show that in every non-geodesic word, there is a sequence of $\tau$-moves that ends with a reduction in the word. Additionally, we provide a recursive method to apply these sequences, so that we obtain an algorithm with quadratic complexity in the length of the input word. Furthermore, we extend our algorithm to a broader and very big class of Artin groups, namely those without $A_3$ or  $B_3$ subdiagrams. These are joint works with Rubén Blasco-García, Rose Morris-Wright, Sarah Rees, and Derek Holt.

Motoko Kato (The University of the Ryukyus)

Acylindrical hyperbolicity of some Artin groups

Artin groups, also called Artin-Tits groups, have been widely studied since their introduction by Tits in 1960s. In particular, Artin groups are important examples in geometric group theory. For various nonpositively curved or negatively curved properties on discrete groups, Artin groups are interesting targets. In this talk, we treat acylindrical hyperbolicity of Artin groups. Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and the actions on them. By developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones. As an application, we see that the centers of such Artin groups are finite, and that actually they are trivial in many cases. Also as a corollary, we see that irreducible Artin groups of infinite type and of type FC are acylindrically hyperbolic. This is a joint work with Shin-ichi Oguni (Ehime University).

Kyoji Saito (Kyoto University)

Semi-infinite Hodge structures and period maps associated with hyperbolic Artin groups of rank 2

As the classical Artin groups of finite type appear as the fundamental groups of regular orbit spaces of finite Weyl groups, some Artin groups of hyperbolic type appear as the fundamental groups of regular orbit spaces of hyperbolic Weyl groups.

In the classical finite type case, the orbit spaces carry the semi-infinite Hodge structure equipped with primitive forms, which lead to period maps (whose nature is yet to be studies) on which the Artin groups are acting. 

In the present talk, I'll report on the semi-infinite Hodge structure and the period map associated with the hyperbolic root systems of rank 2, on which hyperbolic Artin groups of rank 2 are acting.