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We determine a classification of the endomorphisms of the Artin groups of spherical type $B_n$ for $n\ge 5$, and of their quotients by the center., Comment: 27 pages, 6 figures. arXiv admin note: text overlap with arXiv:2406.02484

We determine a classification of the endomorphisms of the Artin group of affine type $\tilde A_n$ for $n\ge 4$., Comment: v3: Corollary 3.6 added; v2: a reference added, 19 pages, 6 figures

We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection of parabolic subgroups in Euclidean Braid groups $A[\tilde{A}_n]$ is again a parabolic subgroup. To that end, we use that the spherical-type Artin group $A[B_{n+1}]$ is isomorphic to $A[\tilde{A}_n] \rtimes \mathbb{Z}$., Comment: 3 pages, 2 figures

In this paper we determine a classification of the endomorphisms of the Artin group $A [D_n]$ of type $D_n$ for $n\ge 6$. In particular we determine its automorphism group and its outer automorphism group. We also determine a classification of the homomorphisms from $A[D_n]$ to the Artin group $A [A_{n-1}]$ of type $A_{n-1}$ and a classification of the homomorphisms from $A[A_{n-1}]$ to $A[D_n]$ for $n\ge 6$. We show that any endomorphism of the quotient $A [D_n] / Z (A [D_n])$ lifts to an endomorphism of $A [D_n]$ for $n \ge 4$. We deduce a classification of the endomorphisms of $A [D_n] / Z (A [D_n])$, we determine the automorphism and outer automorphism groups of $A [D_n] / Z (A [D_n])$, and we show that $A [D_n] / Z (A [D_n])$ is co-Hopfian, for $n \ge 6$. The results are algebraic in nature but the proofs are based on topological arguments (curves on surfaces and mapping class groups).

Graph products of cyclic groups and Coxeter groups are two families of groups that are defined by labeled graphs. The family of Dyer groups contains these both families and gives us a framework to study these groups in a unified way. This paper focuses on the growth series of a Dyer group $D$ with respect to the standard generating set. We give a recursive formula for the growth series of $D$ in terms of the growth series of standard parabolic subgroups. As an application we obtain the rationality of the growth series of a Dyer group. Furthermore, we show that the growth series of $D$ is closely related to the Euler characteristic of $D$., Comment: 14 pages

One can observe that Coxeter groups and right-angled Artin groups share the same solution to the word problem. On the other hand, in his study of reflection subgroups of Coxeter groups Dyer introduces a family of groups, which we call Dyer groups, which contains both, Coxeter groups and right-angled Artin groups. We show that all Dyer groups have this solution to the word problem, we show that a group which admits such a solution belongs to a little more general family of groups that we call quasi-Dyer groups, and we show that this inclusion is strict. Then we show several results on parabolic subgroups in quasi-Dyer groups and in Dyer groups. Notably, we prove that any intersection of parabolic subgroups in a Dyer group of finite type is a parabolic subgroup.

By definition, a group is called narrow if it does not contain a copy of a non-abelian free group. We describe the structure of finite and narrow normal subgroups in Coxeter groups and their automorphism groups., Comment: To appear in Journal of Group Theory

We prove that a parabolic subgroup $P$ contained in another parabolic subgroup $P'$ of an Artin group $A$ is a parabolic subgroup of $P'$. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction $A \to P$ of the inclusion map from a parabolic subgroup $P$ into $A$. This retraction was implicitly constructed in a previous paper by Charney and the second author., Comment: 8 pages. Accepted for publication in Proc. Amer. Math. Soc

Given an Artin group $A_\Gamma$, a common strategy in the study of $A_\Gamma$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_\Gamma$ has a specific property if and only if all "small" parabolic subgroups of $A_\Gamma$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_\Gamma$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_\Gamma$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of $A_\Gamma$ to fixed point properties and to automatic continuity of $A_\Gamma$ using Bass-Serre theory and a generalization of the Deligne complex., Comment: 19 pages

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

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 {$\rho$ f n : R f [VB n ] $\rightarrow$ VTL n (R f)} $\infty$ n=1 and {$\rho$ 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.

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 $\mathcal{TG}r(G)\simeq\mathcal{TG}r(G')$.

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., Comment: 22 pages, 1 figure

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 $\theta \in T$ is of minimal length in its right-coset, $H \theta$, 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 $H$.

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.

Let $\Gamma$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $\Gamma$.Recall that, by a theorem of H{\'e}e and M\"uhlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat \Gamma$.We denote by $\Phi^+$ the set of positive roots of $\Gamma$ and by $\hat \Phi^+$ the set of positive roots of $\hat \Gamma$.Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $\Phi^+$.The action of $G$ on $\Gamma$ induces an action of $G$ on $\Phi^+$, and therefore on $E$.We show that $E^G$ contains a linearly independent family of vectors naturally in one-to-one correspondence with $\hat \Phi^+$ 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.

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 $\Gamma/\mathcal{P}$. We prove that, if $\Gamma/\mathcal{P}$ 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.

Let $N$ be a compact, connected, non-orientable surface of genus $\rho$ with $n$ boundary components, with $\rho \ge 5$ 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 $\varphi: \mathcal{G} \to \mathcal{M} (N)$ is an injective homomorphism, then there exists $f_0 \in \mathcal{M} (N)$ such that $\varphi (g) = f_0 g f_0^{-1}$ for all $g \in \mathcal{G}$. We deduce that the abstract commensurator of $\mathcal{M} (N)$ coincides with $\mathcal{M} (N)$.

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 $\lambda :\mathcal{T}(N) \rightarrow \mathcal{T}(N)$ is a superinjective simplicial map, then there exists a homeomorphism $h : N \rightarrow N$ unique up to isotopy such that $H(\alpha) = \lambda(\alpha)$ for every vertex $\alpha$ in $\mathcal{T}(N)$ where $H=[h]$., Comment: 51 pages, 36 figures

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

We prove that even Artin groups of FC type are poly-free and residually finite.

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 $(V, \langle .,. \rangle, \Pi)$.We assume that $G \subset \GL (V)$, and that $G$ leaves invariant $\Pi$ and $\langle .,. \rangle$. We show that $W^G$ is a Coxeter group, we construct a subset $\tilde \Pi \subset V^G$ so that $(V^G, \langle .,. \rangle, \tilde \Pi)$ 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 $(V^G, \langle .,. \rangle, \tilde \Pi)$.In particular, the latter is faithful. The fact that $W^G$ is a Coxeter group is already known and is due to M\"uhlherr and H\'ee, but also follows directly from the proof of the other results.

We show a simple and easily implementable solution to the word problem for virtual braid groups., Comment: Minor changes and few examples added. To appear in Pacific Journal of Mathematics

The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.

We prove that any standard parabolic subgroup of any Artin group is convex with respect to the standard generating set.

Let $N_{g,n}$ denote the nonorientable surface of genus $g$ with $n$ boundary components and $M(N_{g,n})$ its mapping class group. We obtain an explicit finite presentation of $M(N_{g,n})$ for $n=0,1$ and all $g$ such that $g+n>3$.

The purpose of this paper is to put together a large amount of results on the $K(\pi,1)$ 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.

For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links.

We define the notion of preGarside group slightly lightening the definition of Garside group so that all Artin-Tits groups are preGarside groups. This paper intends to give a first basic study on these groups. Firstly, we introduce the notion of parabolic subgroup, we prove that any preGarside group has a (partial) complemented presentation, and we characterize the parbolic subgroups in terms of these presentations. Afterwards we prove that the amalgamated product of two preGarside groups along a common parabolic subgroup is again a preGarside group. This enables us to define the family of preGarside groups of FC type as the smallest family of preGarside groups that contains the Garside groups and that is closed by amalgamation along parabolic subgroups. Finally, we make an algebraic and combinatorial study on FC type preGarside groups and their parabolic subgroups.

Let $g\geq3$ and $n\geq0$, and let ${\mathcal{M}}_{g,n}$ be the mapping class group of a surface of genus $g$ with $n$ boundary components. We prove that ${\mathcal{M}}_{g,n}$ contains a unique subgroup of index $2^{g-1}(2^{g}-1)$ up to conjugation, a unique subgroup of index $2^{g-1}(2^{g}+1)$ up to conjugation, and the other proper subgroups of ${\mathcal{M}}_{g,n}$ are of index greater than $2^{g-1}(2^{g}+1)$. In particular, the minimum index for a proper subgroup of ${\mathcal{M}}_{g,n}$ is $2^{g-1}(2^{g}-1)$.

This paper is a short survey on four basic questions on Artin-Tits groups: the torsion, the center, the word problem, and the cohomology ($K(\pi,1)$ problem). It is also an opportunity to prove three new results concerning these questions: (1) if all free of infinity Artin-Tits groups are torsion free, then all Artin-Tits groups will be torsion free; (2) If all free of infinity irreducible non-spherical type Artin-Tits groups have a trivial center then all irreducible non-spherical type Artin-Tits groups will have a trivial center; (3) if all free of infinity Artin-Tits groups have solutions to the word problem, then all Artin-Tits groups will have solutions to the word problem. Recall that an Artin-Tits group is free of infinity if its Coxeter graph has no edge labeled by $\infty$.

We describe a new algorithm which for each braid returns a quasi-geodesic sigma-definite word representative, defined as a braid word in which the generator sigma_i with maximal index i appears either only positively or only negatively., Comment: 11 pages

Let $\Gamma$ be a Coxeter graph, let $(W,S)$ be its associated Coxeter system, and let $(A,\Sigma$) be its associated Artin-Tits system. We regard $W$ as a reflection group acting on a real vector space $V$. Let $I$ be the Tits cone, and let $E_\Gamma$ be the complement in $I +iV$ of the reflecting hyperplanes. Recall that Charney, Davis, and Salvetti have constructed a simplicial complex $\Omega(\Gamma)$ having the same homotopy type as $E_\Gamma$. We observe that, if $T \subset S$, then $\Omega(\Gamma_T)$ naturally embeds into $\Omega (\Gamma)$. We prove that this embedding admits a retraction $\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)$, and we deduce several topological and combinatorial results on parabolic subgroups of $A$. From a family $\SS$ of subsets of $S$ having certain properties, we construct a cube complex $\Phi$, we show that $\Phi$ has the same homotopy type as the universal cover of $E_\Gamma$, and we prove that $\Phi$ is CAT(0) if and only if $\SS$ is a flag complex. We say that $X \subset S$ is free of infinity if $\Gamma_X$ has no edge labeled by $\infty$. We show that, if $E_{\Gamma_X}$ is aspherical and $A_X$ has a solution to the word problem for all $X \subset S$ free of infinity, then $E_\Gamma$ is aspherical and $A$ has a solution to the word problem. We apply these results to the virtual braid group $VB_n$. In particular, we give a solution to the word problem in $VB_n$, and we prove that the virtual cohomological dimension of $VB_n$ is $n-1$.

This paper is a presentation, where we compute the HOMFLYPT Skein module of singular links in the 3-sphere. This calculation is based on some results previously proved by Rabenda and the author on Markov traces on singular Hecke algebras, as well as on classical techniques that allow to pass from the framework of Markov traces on Hecke algebras to the framework of HOMFLYPT Skein modules. Some open problems on singular Hecke algebras are also presented.

We prove that the mapping class group of a closed oriented surface of genus $\rho \ge 3$ has no proper subgroup of index $\le 4 \rho +4$.

This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of the theory: the faithful linear representations, the cohomology, and the geometrical representations.

We define the singular Hecke algebra ${\mathcal H} (SB_n)$ as the quotient of the singular braid monoid algebra ${\mathbb C} (q) [SB_n]$ by the Hecke relations $\sigma_k^2 = (q-1) \sigma_k +q$, $1 \le k\le n-1$, and define the Markov traces on the sequence $\{{\mathcal H}(SB_n)\}_{n=1}^{+\infty}$ in the same way as for the Markov traces on the tower of (non-singular) Hecke algebras of the symmetric groups. We prove that the Markov traces are in one-to-one correspondance with the invariants that satisfies some skein relation, and compute an explicit classification of the Markov traces. Thanks to this classification, we define some universal HOMFLY-type invariant which has the property that it distinguishes all the pairs of singular links that can be distinguished by an invariant which satisfies the required skein relation.

Let $\mathcal M (\Sigma, \mathcal P)$ be the mapping class group of a punctured oriented surface $(\Sigma, \mathcal P)$ (where $\mathcal P$ may be empty), and let $\mathcal T_p(\Sigma,\mathcal P)$ be the kernel of the action of $\mathcal M (\Sigma, \mathcal P)$ on $H_1 (\Sigma \setminus \mathcal P, \mathbb F_p)$. We prove that $\mathcal T_p(\Sigma, \mathcal P)$ is residually $p$. In particular, this shows that $\mathcal M (\Sigma, \mathcal P)$ is virtually residually $p$. For a group $G$ we denote by $\mathcal I_p(G)$ the kernel of the natural action of ${\rm Out} (G)$ on $H_1(G, \mathbb F_p)$. In order to achieve our theorem, we prove that, under certain conditions ($G$ is conjugacy $p$-separable and has Property A), the group $\mathcal I_p(G)$ is residually $p$. The fact that free groups and surface groups have Property A is due to Grossman. The fact that free groups are conjugacy $p$-separable is due to Lyndon and Schupp. The fact that surface groups are conjugacy $p$-separable is, from a technical point of view, the main result of the paper.

The purpose of this paper is the study of the roots in the mapping class groups. Let $\Sigma$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $\Sigma$, and let $\MM (\Sigma, \PP)$ denote the mapping class group of $(\Sigma, \PP)$. We prove that, if $\Sigma$ is of genus 0, then each $f \in \MM (\Sigma)$ has at most one $m$-root for all $m \ge 1$. We prove that, if $\Sigma$ is of genus 1 and has non-empty boundary, then each $f \in \MM (\Sigma)$ has at most one $m$-root up to conjugation for all $m \ge 1$. We prove that, however, if $\Sigma$ is of genus $\ge 2$, then there exist $f,g \in \MM (\Sigma, \PP)$ such that $f^2=g^2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Afterwards, we focus our study on the roots of the pseudo-Anosov elements. We prove that, if $\partial \Sigma \neq \emptyset$, then each pseudo-Anosov element $f \in \MM(\Sigma, \PP)$ has at most one $m$-root for all $m \ge 1$. We prove that, however, if $\partial \Sigma = \emptyset$ and the genus of $\Sigma$ is $\ge 2$, then there exist two pseudo-Anosov elements $f,g \in \MM (\Sigma)$ (explicitely constructed) such that $f^m=g^m$ for some $m\ge 2$, $f$ is not conjugate to $g$, and none of the conjugates of $f$ commutes with $g$. Furthermore, if the genus of $\Sigma$ is $\equiv 0 (\mod 4)$, then we can take $m=2$. Finally, we show that, if $\Gamma$ is a pure subgroup of $\MM (\Sigma, \PP)$ and $f \in \Gamma$, then $f$ has at most one $m$-root in $\Gamma$ for all $m \ge 1$. Note that there are finite index pure subgroups in $\MM (\Sigma, \PP)$.

We prove that a non-spherical irreducible Coxeter group is (directly) indecomposable and that a non-spherical and non-affine Coxeter group is strongly indecomposable in the sense that all its finite index subgroups are (directly) indecomposable. We prove that a Coxeter group has a decomposition as a direct product of indecomposable groups, and that such a decomposition is unique up to a central automorphism and a permutation of the factors. We prove that a Coxeter group has a virtual decomposition as a direct product of strongly indecomposable groups, and that such a decomposition is unique up to commensurability and a permutation of the factors.

This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.