Braids and beyond / De plectis prospectibusque
Conference in memory of Patrick Dehornoy
Caen, September 8-10 2021, hybrid mode

While infinite type Artin groups present many challenges, the corresponding Artin monoids are generally easier to understand using aspects of Garside theory. I will talk about some joint work with Rachael Boyd and Rose Morris-Wright on Deligne complexes for Artin monoids and their relation to the Deligne complex for the Artin group. This leads to some interesting questions about the Cayley graph of an Artin group with generating set the full monoid, and implications of Dehornoy's "multifraction reduction" conditions.
(Joint work with Rachael Boyd and Rose Morris-Wright.)
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There is a generalization of Thompson's groups constructed from the Thompson's group $V$ and Artin's braid group: the braided Thompson's group $BV_2$, which was independently introduced by Patrick Dehornoy and Matthew G. Brin in 2006. In this talk we will explain how to extend this concept to a much bigger family of groups by using infinite braids: Infinitely braided Thompson's groups $BV_n(H)$, where $H$ is a subgroup of the braid group on $n$ strands. We will prove that they are indeed groups by using braided diagrams and rewriting systems. We will also see that they are finitely generated if $H$ is finitely generated and give an explicit set of generators for $BV_n(H)$ and some other cases.
(Joint work with Julio Aroca.)
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Garside families have been introduced by Patrick Dehornoy and his coauthors in an endeavor to find the right abstractions behind the nice normal form properties of Artin–Tits monoids, and of Garside monoids. Deligne had constructed coherent presentations for spherical (i.e. with finite associated Coxeter group) Artin–Tits monoids. Here, "coherent" qualifies a presentation enhanced with a collection of relations among relations that "fill" any parallel pair of sequences of relations applied in context between two given words. Gaussent, Guiraud and Malbos have extended this result to all Artin–Tits monoids on one hand, and to Garside monoids on the other hand. In this work, we unify and generalise these two results by presenting coherent presentations in the framework of Garside families.
(Joint work with Alen Đurić and Yves Guiraud.)
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A few years ago, in a private communication, Oleg Viro proposed, without proof, an analogue of Markov's theorem and Alexander's lemma for links in the three-dimensional projective space. Any oriented link in the projective space can be presented as the closure of a braid, and two braids representing equivalent links are related by a sequence of moves that resemble the classical Markov moves. Jointly with Stepan Orevkov we found a simple proof of this fact. The claim can be reformulated in a way that is naturally generalized to any lense space, and the proof is generalized accordingly. The work is supported by the Russian Science Foundation under grant no. 19-11-00151.
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We introduce a new algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. Our experimentations in the case of three and four strands allow us to conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.
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Motivated by a question of Dehornoy–Digne–Godelle–Krammer–Michel on a submonoid of Artin's braid group, we give a new Garside structure for torus knot groups. In a specific case, these groups are extensions of Artin's braid group, and the new Garside monoid surjects onto DDGKM's monoid. This allows one to deduce a few properties of the latter, to partially answer the above mentioned question, and also to give a conjectural finite presentation of DDGKM's monoid. In the second part of the talk, we give some evidence that torus knot groups may be considered as braid groups of a family of (in general infinite) generalizations of complex reflection groups naturally obtained as quotients. These "reflection" groups have a nontrivial center, and we show that the quotient by their center is isomorphic to the alternating subgroup of a Coxeter group of rank three.

We present a method, inspired by techniques developed by Patrick Dehornoy, to deal with parabolic subgroups in certain Garside groups. We show that, under some hypotheses, every element in a Garside group admits a parabolic closure, that is, a unique parabolic subgroup, minimal by inclusion, containing the element. These hypotheses are satisfied by Artin–Tits groups of spherical type and by some complex braid groups.
(Joint work with Ivan Marin.)
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We start with a quick general survey of some of the results that have been proved for various classes of Artin groups (also known as Artin–Tits groups), with emphasis on those that involve the word problem. We go on to summarise very briefly a project initiated by Patrick Dehornoy in 2016 involving the use of multifraction reduction in solving the word problem in various classes of Artin groups, and his ambitious conjecture that this method could be applied successfully to on all Artin groups. We also mention some joint work of Patrick with Sarah Rees and myself on the application of this method to large-type Artin groups. We conclude with a brief description of some recently proven isomorphisms between quotients of various Artin groups that arise as interval groups associated with quasi-Coxeter elements in finite Coxeter groups.
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A few years ago I investigated a geometrically-defined action of the braid group $B_{2n+2}$ on the free group on $2n$ generators. Linearizing this action, one obtains a group homomorphism from $B_{2n+2}$ to the symplectic modular group $Sp_{2n}({\mathbb Z})$. With François Digne we showed that this homomorphism lifts to the integral Steinberg group $St(C_n,{\mathbb Z})$ of type $C_n$. When $n=1$ or $n=2$, we know the image and the kernel of the homomorphism. I will also report on partial results in the general case $n\geq 3$.
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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 (at Patrick's retirement conference), 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.
(Joint work with Michael Dougherty.)
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As the title says, with François Digne we have been trying to understand and make simpler if possible the remarkable work of the above authors. I will present some simplifications, like a unified treatment of the ribbon category, and a new summit set, the reversing circuits, which is not optimized for size but is optimized for making proofs simple.
(Joint work with François Digne.)
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Starting from the observation that the standard presentation of a virtual braid group mixes the presentations of the corresponding braid group and of the corresponding symmetric group together with the action of the symmetric group on its root system, we define a virtual Artin group ${\rm VA}[\Gamma]$ with a presentation that mixes the standard presentations of the Artin group $A[\Gamma]$ and of the Coxeter group $W[\Gamma]$ together with the action of $W[\Gamma]$ on its root system. By definition we have two epimorphisms $\pi_K:{\rm VA}[\Gamma]\to W[\Gamma]$ and $\pi_P:{\rm VA}[\Gamma]\to W[\Gamma]$ whose kernels are denoted by ${\rm KVA}[\Gamma]$ and ${\rm PVA}[\Gamma]$, respectively. In this talk we will focus on ${\rm KVA}[\Gamma]$. We will show that this group is an Artin group whose standard generating set is in one-to-one correspondence with the root system of $W[\Gamma]$. Afterwards, we use this presentation to show that the center of ${\rm VA}[\Gamma]$ is always trivial, and to show that ${\rm VA}[\Gamma]$ has a solvable word problem when $\Gamma$ is of spherical type or of affine type.
(Joint work with Paolo Bellingeri and Anne-Laure Thiel.)
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Via the Artin representation, an $n$-braid corresponds to an automorphism of the free group $F_n$. Call a braid "order-preserving" if the corresponding automorphism preserves some bi-ordeing of $F_n$. This is equivalent to bi-orderability of the fundamental group of the complement of the link consisting of the union of the braid closure and the braid axis. We apply this to orderability of fundamental groups of minimal volume cusped hyperbolic $3$-manifolds. For example, of the two minimal-volume one-cusped hyperbolic $3$-manifolds, one has bi-orderable fundamental group whereas the other does not.
(Joint work with Eiko Kin.)
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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 corollary, we prove that Morse elements act loxodromically on the additional length complex of $G$.
(Joint work with Matthieu Calvez.)
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