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Luis Paris

Publications and source records attributed to Luis Paris.

At least 37 records · Page 2Linked to original sources
arXiv2021

Virtual Artin groups

Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group $VA[Γ]$ of a Coxeter graph $Γ$ mixing the standard presentation of the Artin group $A[Γ]$ with the standard presentation of the Coxeter group $W[Γ]$ and some mixed relations that mimic the action of $W[Γ]$ on its root system. By definition we have two epimorphisms $π_K:VA[Γ]\to W[Γ]$ and $π_P:VA[Γ]\to W[Γ]$ whose kernels are denoted by $KVA[Γ]$ and $PVA[Γ]$ respectively. We calculate presentations for these two subgroups. In particular $KVA[Γ]$ is an Artin group. We prove that the center of any virtual Artin group is trivial. In the case where $Γ$ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of $KVA[Γ]$ is also of spherical type or of affine type, and we show that $VA[Γ]$ has a solution to the word problem. In the case where $Γ$ is of spherical type we show that $KVA[Γ]$ satisfies the $K(π,1)$ conjecture and we infer the cohomological dimension of $KVA[Γ]$ and the virtual cohomological dimension of $VA[Γ]$. In the case where $Γ$ is of affine type we determine upper bounds for the cohomological dimension of $KVA[Γ]$ and for the virtual cohomological dimension of $VA[Γ]$.

math.GR↗
arXiv2020

Commensurability in Artin groups of spherical type

Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for every $i$. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed $n$, we give a complete classification of the irreducible Artin groups of rank $n$ that are commensurable with the group of type $A_n$. Note that it will remain 6 pairs of groups to compare to get the complete classification of Artin groups of spherical type up to commensurability.

math.GR↗
arXiv2020

Virtual an arrow Temperley--Lieb algebras, Markov traces, and virtual link invariants

Let R f = Z[A $\pm$1 ] be the algebra of Laurent polynomials in the variable A and let R a = Z[A $\pm$1 , z 1 , z 2 ,. .. ] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z 1 , z 2 ,. .. . For n $\ge$ 1 we denote by VB n the virtual braid group on n strands. We define two towers of algebras {VTL n (R f)} $\infty$ n=1 and {ATL n (R a)} $\infty$ n=1 in terms of diagrams. For each n $\ge$ 1 we determine presentations for both, VTL n (R f) and ATL n (R a). We determine sequences of homomorphisms {$ρ$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\infty$ n=1 and {$ρ$ a n : R a [VB n ] $\rightarrow$ ATL n (R a)} $\infty$ n=1 , we determine Markov traces {T f n : VTL n (R f) $\rightarrow$ R f } $\infty$ n=1 and {T a n : ATL n (R a) $\rightarrow$ R a } $\infty$ n=1 , and we show that the invariants for virtual links obtained from these Markov traces are the f-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each n $\ge$ 1, the standard Temperley-Lieb algebra TL n embeds into both, VTL n (R f) and ATL n (R a), and that the restrictions to {TL n } $\infty$ n=1 of the two Markov traces coincide.

math.GR↗
arXiv2019

On the isomorphism problem for even Artin groups

An even Artin group is a group which has a presentation with relations of the form $(st)^n=(ts)^n$ with $n\ge 1$. With a group $G$ we associate a Lie $\mathbb Z$-algebra $\mathcal{TG}r(G)$. This is the usual Lie algebra defined from the lower central series, truncated at the third rank. For each even Artin group $G$ we determine a presentation for $\mathcal{TG}r(G)$. Then we prove a criterion to determine whether two Coxeter matrices are isomorphic. Let $c,d\in\mathbb N$ such that $c\ge1$, $d\ge2$ and $\gcd(c,d)=1$. We show that, if two even Artin groups $G$ and $G'$ having presentations with relations of the form $(st)^n=(ts)^n$ with $n\in\{c\}\cup\{d^k\mid k\ge1\}$ are such that $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$, then $G$ and $G'$ have the same presentation up to permutation of the generators. On the other hand, we show an example of two non-isomorphic even Artin groups $G$ and $G'$ such that $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$.

math.GR↗
arXiv2019

Transverse properties of parabolic subgroups of Garside groups

Let $G$ be a Garside group endowed with the generating set $\mathcal{S}$ of non-trivial simple elements, and let $H$ be a parabolic subgroup of $G$. We determine a transversal $T$ of $H$ in $G$ such that each $θ\in T$ is of minimal length in its right-coset, $H θ$, for the word length with respect to $\mathcal{S}$. We show that there exists a regular language $L$ on $\mathcal{S} \cup \mathcal{S}^{-1}$ and a bijection $\mathrm{ev} : L \to T$ satisfying $\mathrm{lg} (U) = \mathrm{lg}_\mathcal{S}( \mathrm{ev}(U))$ for all $U \in L$. From this we deduce that the coset growth series of $H$ in $G$ is rational. Finally, we show that $G$ has fellow projections on $H$ but does not have bounded projections on $H$.

math.GR↗
arXiv2018

Virtual braids and permutations

Let VB$_n$ be the virtual braid group on $n$ strands and let $\mathfrak{S}_n$ be the symmetric group on $n$ letters. Let $n,m \in \mathbb{N}$ such that $n \ge 5$, $m \ge 2$ and $n \ge m$. We determine all possible homomorphisms from VB$_n$ to $\mathfrak{S}_m$, from $\mathfrak{S}_n$ to VB$_m$ and from VB$_n$ to VB$_m$. As corollaries we get that Out(VB$_n$) is isomorphic to $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and that VB$_n$ is both Hopfian and co-Hofpian.

math.GR↗
arXiv2018

Ordering Garside groups

We introduce a condition on Garside groups that we call Dehornoy structure. An iteration of such a structure leads to a left order on the group. We show conditions for a Garside group to admit a Dehornoy structure, and we apply these criteria to prove that the Artin groups of type A and I 2 (m), m $\ge$ 4, have Dehornoy structures. We show that the left orders on the Artin groups of type A obtained from their Dehornoy structures are the Dehornoy orders. In the case of the Artin groups of type I 2 (m), m $\ge$ 4, we show that the left orders derived from their Dehornoy structures coincide with the orders obtained from embeddings of the groups into braid groups. 20F36

math.GR↗
arXiv2018

Root systems, symmetries and linear representations of Artin groups

Let $Γ$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $Γ$.Recall that, by a theorem of H{é}e and Mühlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat Γ$.We denote by $Φ^+$ the set of positive roots of $Γ$ and by $\hat Φ^+$ the set of positive roots of $\hat Γ$.Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $Φ^+$.The action of $G$ on $Γ$ induces an action of $G$ on $Φ^+$, and therefore on $E$.We show that $E^G$ contains a linearly independent family of vectors naturally in one-to-one correspondence with $\hat Φ^+$ and we determine exactly when this family is a basis of $E^G$.This question is motivated by the construction of Krammer's style linear representations for non simply laced Artin groups.

math.GR↗
arXiv2017

Note on residual finiteness of Artin groups

Let $A$ be an Artin group. A partition $\mathcal{P}$ of the set of standard generators of $A$ is called admissible if, for all $X,Y \in \mathcal{P}$, $X \neq Y$, there is at most one pair $(s,t) \in X \times Y$ which has a relation. An admissible partition $\mathcal{P}$ determines a quotient Coxeter graph $Γ/\mathcal{P}$. We prove that, if $Γ/\mathcal{P}$ is either a forest or an even triangle free Coxeter graph and $A_X$ is residually finite for all $X \in \mathcal{P}$, then $A$ is residually finite.

math.GR↗
arXiv2017

Injective homomorphisms of mapping class groups of non-orientable surfaces

Let $N$ be a compact, connected, non-orientable surface of genus $ρ$ with $n$ boundary components, with $ρ\ge 5$ and $n \ge 0$, and let $\mathcal{M} (N)$ be the mapping class group of $N$. We show that, if $\mathcal{G}$ is a finite index subgroup of $\mathcal{M} (N)$ and $φ: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $φ(g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.

math.GT↗
arXiv2017

Superinjective Simplicial Maps of the Two-sided Curve Complexes on Nonorientable Surfaces

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components with $g \geq 5$, $n \geq 0$. Let $\mathcal{T}(N)$ be the two-sided curve complex of $N$. If $λ:\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(α) = λ(α)$ for every vertex $α$ in $\mathcal{T}(N)$ where $H=[h]$.

math.GT↗
arXiv2016

Coxeter groups, symmetries, and rooted representations

Let $(W,S)$ be a Coxeter system, let $G$ be a group of symmetries of $(W,S)$ and let $f : W \to \GL (V)$ be the linear representation associated with a root basis $(V, \langle .,. \rangle, Π)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $Π$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde Π\subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde Π)$ is a root basis of $W^G$, and we show that the induced representation $f^G : W^G \to \GL(V^G)$ is the linear representation associated with $(V^G, \langle .,. \rangle, \tilde Π)$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to Mühlherr and Hée, but also follows directly from the proof of the other results.

math.GR↗
arXiv2012

$K(π,1)$ conjecture for Artin groups

The purpose of this paper is to put together a large amount of results on the $K(π,1)$ conjecture for Artin groups, and to make them accessible to non-experts. Firstly, this is a survey, containing basic definitions, the main results, examples and an historical overview of the subject. But, it is also a reference text on the topic that contains proofs of a large part of the results on this question. Some proofs as well as few results are new. Furthermore, the text, being addressed to non-experts, is as self-contained as possible.

math.GR↗