Presentation of pure braid groups

François Digne and Yasushi Gomi

0. Introduction. In this article we study by combinatorial methods presentations of the pure braid group associated with any Coxeter group. We generalize to all types the decomposition of the pure braid group into successive semi-direct products (known in the case of type A_{n}, cf. for example [Bi, 1.8.2] or [H, appendix by Gædde]). Some results of the same type can be found in [BrMaRo, section 3].

Let us give a review of our results. In section 2 we get a presentation of the subgroup D_{I} of a braid group which maps onto a given parabolic subgroup W_{I} of the associated Coxeter group W. We recover the fact that the braid group B_{I} of type W_{I} imbeds into the braid group of type W and we show that D_{I} has a semi-direct decomposition with quotient B_{I}. The most interesting case is when the parabolic subgroup W_{I} is maximal of the same type as W. We study this case in section 3 for all infinite series of finite Coxeter groups. As a consequence we get a simple presentation of the pure braid group of type B_{n} which, as in type A_{n}, appears to be an n-fold semi-direct product of free groups. In Section 4, we show how the action of the braid group of type B_{n} on a free group can be used to prove the embedding of this group into the braid group of type A_{n}.

1. Notation and preliminaries. Throughout this paper we denote by (W, S) a Coxeter system and by B_{W}^{+} (resp. B_{W}) the corresponding braid monoid (resp. braid group). We review the definitions. For s and t in S let m_{s,t} ∈ ℕ ∪ {∞} be the order of st in W and S a set in one-to-one correspondence with S, the bijection being denoted by s ↦ s. By definition B_{W}^{+} (resp. B_{W}) is the monoid (resp. the group) generated by S with defining relations \underbrace{st …}_{m_{s,t}} = \underbrace{ts …}_{m_{s,t}}, for all s ∈ S and all t ∈ S.

François Digne and Yasushi Gomi

We have then a canonical homomorphism p from B_{W} to W which maps s to s. The pure braid group is by definition the kernel of p and will be denoted by P_{W}. We also have a canonical morphism j : B_{W}^{+} → B_{W}. This morphism is known to be injective in some cases, in particular when W is finite and is conjectured to be so in all cases. The surjection p ∘ j has a canonical section q, mapping s on s and mapping an arbitrary element w ∈ W on the pull back of any of its reduced expressions (this is independent on the choice of the reduced expression). We denote by B_{W}^{red} the image of the section q in B_{W}^{+} , or in B_{W} (note that j induces a bijection between these two sets). We denote by w the image in B_{W}^{+} or in B_{W} of w ∈ W by q. The set B_{W}^{red} consists of the elements whose length with respect to S is equal to the length of their image in W with respect to S.

If b is an element of B_{W} of the form b = s_{1}^{ε_{1}}s_{2}^{ε_{2}} … s_{k}^{ε_{k}}, with s_{i} ∈ S and ε_{i} = ±1, we put b̃ = s_{k}^{ε_{k}} … s_{2}^{ε_{2}}s_{1}^{ε_{1}}. This element is well-defined, independent on the decomposition of b as a product of generators since the defining relations of B_{W} are invariant by reversing the words. If I is a subset of S we write W_{I} for the parabolic subgroup of W generated by I. If I is a spherical subset of S (that is, if W_{I} is finite), we denote by w_{I} the longest element in the parabolic subgroup W_{I}, we denote by I the subset of S corresponding to I and put w_{I} = q(w_{I}). We will often use the next lemma which is a version adapted to braid monoids of the exchange lemma.

Lemma 1.1. Let s, t ∈ S and b ∈ B_{W}^{red}; if sb ∈ B_{W}^{red} and bt ∈ B_{W}^{red} but sbt ∉ B_{W}^{red} then sb = bt.

2. Presentations of certain subgroups of the braid groups. We apply the Reidemeister-Schreier method ([J, chapter 9]) to obtain presentations of some subgroups of B_{W} and in particular of the pure braid group. Let I be a subset of S. We denote by D_{I} the inverse image of W_{I} in B_{W}. This subgroup contains the pure braid group and the parabolic subgroup B_{I} of B_{W} where we denote I the subset of S whose image in W is I. We say that an element w ∈ B_{W}^{red} is I-reduced if we have sw ∈ B_{W}^{red} for all s ∈ I. This is equivalent to the image w of w in W being I-reduced, that is, the length of w being minimal in the right coset W_{I}w. In a similar way, we have the notion of reduced-I elements. For I, J ⊂ S, if an element is both I-reduced and reduced-J we shall say that it is I-reduced-J.

We now find a presentation of D_{I} by applying the Reidemeister-Schreier method. The set of I-reduced elements of B_{W}^{red} forms a set of representatives of the quotient D_{I}\B_{W} since D_{I}\B_{W} ≃ W_{I}\W. We denote by [b] the representative of the class D_{I}b. The elements ws[ws]^{−1} where w ranges over the set of I-reduced elements and where s ranges over S form a set of generators of D_{I}. If ws is in B_{W}^{red} and is I-reduced, the obtained generator equals 1. If ws is in B_{W}^{red} but is not I-reduced, then there exists t ∈ I such that ws = t[ws]. The obtained generator equals t; this includes the case where w = 1, hence t ranges over the whole set I. If ws is not in B_{W}^{red} then w = vs, the element v is I-reduced-{s} and is equal to [ws] since ws = vs^{2}. Then the obtained generator equals vs^{2}v^{−1}. Hence D_{I} is generated by I and by the elements ws^{2}w^{−1} where s is in S and ws ∈ B_{W}^{red} is I-reduced.

Pure braid groups

Definition 2.1. For w ∈ B_{W}^{red} and s ∈ S such that ws ∈ B_{W}^{red} is I-reduced, we put a_{w,s} = ws^{2}w^{−1}.

The Reidemeister-Schreier method gives, for each representative of D_{I}\B_{W} and each braid relation between two elements of S, a relation between the generators of D_{I} through the rewriting process, the relations thus obtained being a complete set of relations.

More precisely for each representative b, that is, each I-reduced element of B_{W}^{red}, and each pair (s, t) ∈ S × S, such that m_{s,t} is finite, we rewrite the relation

b \underbrace{st …}_{m_{s,t}} = b \underbrace{ts …}_{m_{s,t}} .  (∗)

Let us review the rewriting process: to each I-reduced element b ∈ B_{W}^{red} and each element s ∈ S we have associated above a generator bs[bs]^{−1} of D_{I}. The rewriting process consists in replacing the product bs by (bs[bs]^{−1})[bs]; doing this process repeatedly, we can express any element of B_{W} as a product of generators of D_{I} and a representative of D_{I}\B_{W}.

Applying this rewriting process to both sides of (∗) we obtain a relation between the generators of D_{I} and when we vary the elements b, s and t we obtain a complete set of relations.

In order to do the computation we use the following notation.

Notation 2.2. A product of i ≤ m_{s,t} factors alternatively equal to s and t and beginning with s is denoted by < s, t >^{(i)}.

Now we fix s, t ∈ S and b ∈ B_{W}^{red} such that m_{s,t} is finite and b is I-reduced. For convenience we write m instead of m_{s,t}. Exchanging s and t if required, we can write b = b_{0}.< s, t >^{(i)} with 0 ≤ i ≤ m and i maximal, that is, b_{0} is reduced-{s, t}. For 0 ≤ j < m, we write a_{s,t}^{(j)} instead of a_{b_{0}<s,t>^{(j)},r} where r is equal to s if j is even and t otherwise (that is, r is the last term of < s, t >^{(j+1)}).

We divide the calculation into the following cases.

Case 1): We have i = 0.

Case 2): We have i ≥ 1 and b_{0}t is I-reduced; in that case by assumption b_{0}s is I-reduced, since i ≥ 1.

Case 3): We have i ≥ 1 and b_{0}t = s′b_{0} with s′ ∈ I; in that case too b_{0}s is I-reduced. To apply the rewriting process in the above three cases, we need the following lemma,

the first part of which is well-known.

Lemma 2.3. Let s and t be two elements of S such that m_{s,t} is finite.

(i) If b ∈ B_{W}^{red} is such that bs and bt are in B_{W}^{red}, then bw_{{s,t}} ∈ B_{W}^{red}. (ii) If b ∈ B_{W}^{red} is a reduced-{s, t} element and if bs and bt are I-reduced, then bw_{{s,t}}

is I-reduced.

Proof of (ii): If bw_{{s,t}} is not I-reduced, let i be the smallest index such that b.< s, t >^{(i)} is not I-reduced. We have i > 1 and by 1.1 there exists an element s′ ∈ I such that b.< s, t >^{(i)} = s′b.< s, t >^{(i−1)}. This equation implies that b.< s, t >^{(i)} is right divisible by s and t (since i − 1 > 0), hence is divisible by w_{{s,t}}, this implies i = m_{s,t} since b is reduced-{s, t}. Then we can divide it by < s, t >^{(i−1)} on the right and we obtain that s′b is equal to bs or bt (depending on the value of i), which contradicts the hypothesis.

We now turn to the rewriting of relation (∗).

Case 1). When i = 0, if b_{0}s and b_{0}t are both I-reduced, then the obtained relation is trivial 1 = 1. If b_{0}s = s′b_{0} with s′ ∈ I and if b_{0}t is I-reduced, then b_{0}< t, s >^{(m−1)} must be I-reduced by the argument of 2.3(ii), and we obtain the trivial relation s′ = s′. If b_{0}s = s′b_{0} and b_{0}t = t′b_{0} with s′ and t′ in I, we obtain the braid relation between s′ and t′ (remark that m_{s,t} = m_{s′,t′} in this case). From this last case we obtain all the braid relations between the elements of I since it contains in particular the case where b_{0} = 1.

Case 2). When i ≥ 1 and b_{0}t is I-reduced, on one side of (∗), depending on the parity of i, the rewriting process is trivial until we get to b_{0}w_{{s,t}}. The next step applied to the product of b_{0}w_{{s,t}} by s or t gives a_{t,s}^{(m−1)}b_{0}< t, s >^{(m−1)}. We continue the rewriting process and we obtain the form

a_{t,s}^{(m−1)}a_{t,s}^{(m−2)} … a_{t,s}^{(m−i)}b_{0}< t, s >^{(m−i)}.

On the other side the same type of calculation gives

a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}b_{0}< t, s >^{(m−i)}.

Hence the obtained relation is

a_{t,s}^{(m−1)}a_{t,s}^{(m−2)} … a_{t,s}^{(m−i)} = a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}.

Case 3). When i ≥ 1 and b_{0}t = s′b_{0} with s′ ∈ I, on one side of (∗), as in the previous case, the rewriting process is trivial until we get to b_{0}w_{{s,t}} = s′b_{0}< s, t >^{(m−1)}. We continue the rewriting and we obtain

s′a_{s,t}^{(m−2)}a_{s,t}^{(m−3)} … a_{s,t}^{(m−i−1)}b_{0}< s, t >^{(m−i−1)}.

On the other side the first steps of the rewriting process go on as in the above case, until it gets to a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}b_{0}t = a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}s′b_{0}. The following steps are trivial and we obtain

a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}s′b_{0}< s, t >^{(m−i−1)}.

Hence the obtained relation is

s′a_{s,t}^{(m−2)}a_{s,t}^{(m−3)} … a_{s,t}^{(m−i−1)} = a_{s,t}^{(i−1)}a_{s,t}^{(i−2)} … a_{s,t}^{(0)}s′.

In order to get a complete set of relations we must vary b_{0} in the above computations: we write a_{b_{0},s,t}^{(i)} instead of a_{s,t}^{(i)}, that is, a_{b_{0},s,t}^{(i)} = a_{b_{0}<s,t>^{(i)},r}, with notation as in 2.1, where r equals s or t depending on i being even or odd. We have proved:

Proposition 2.4. The group D_{I} has a presentation where the generators are the elements of I and the a_{b,s}, where s ∈ S and bs ∈ B_{W}^{red} is I-reduced, and where the relations are the braid relations between the elements of I and the relations

a_{b_{0},s,t}^{(m−1)}a_{b_{0},s,t}^{(m−2)} … a_{b_{0},s,t}^{(m−i)} = a_{b_{0},t,s}^{(i−1)}a_{b_{0},t,s}^{(i−2)} … a_{b_{0},t,s}^{(0)} for i = 1, …, m  (1)
s′a_{b_{0},s,t}^{(m−2)}a_{b_{0},s,t}^{(m−3)} … a_{b_{0},s,t}^{(m−i−1)} = a_{b_{0},s,t}^{(i−1)}a_{b_{0},s,t}^{(i−2)} … a_{b_{0},s,t}^{(0)}s′ for i = 1, …, m − 1, (2)

where (s, t) runs over all elements of S × S such that m = m_{s,t} is finite, where in (1) b_{0} ∈ B_{W}^{red} runs over the set of reduced-{s, t} elements such that b_{0}s and b_{0}t are I-reduced and in (2) b_{0} ∈ B_{W}^{red} runs over the set of reduced-{s, t} elements such that b_{0}s is I-reduced and that b_{0}t = s′b_{0} with s′ ∈ I.

In particular the case of the pure braid group is given by the next result.

Corollary 2.5. The group P_{W} has a presentation where the generators are the a_{b,s} where s ∈ S and bs ∈ B_{W}^{red} and where the relations are, with the notation of proposition 2.4,

a_{b_{0},s,t}^{(m−1)}a_{b_{0},s,t}^{(m−2)} … a_{b_{0},s,t}^{(m−i)} = a_{b_{0},t,s}^{(i−1)}a_{b_{0},t,s}^{(i−2)} … a_{b_{0},t,s}^{(0)} for i = 1, …, m,

for each pair (s, t) ∈ S × S such that m_{s,t} is finite and for each reduced-{s, t} element b_{0} ∈ B_{W}^{red}.

Proof: the result is immediate by applying proposition 2.4 to the case where I is empty.

Corollary 2.6.

(i) The group B_{I} is isomorphic to the braid group B_{W_{I}}.

(ii) If U_{I} is the normal subgroup of D_{I} generated by the elements a_{b,s} as in 2.4 then D_{I} = U_{I} ⋊ B_{I}.

Proof: We can define a homomorphism h : D_{I} → B_{W_{I}} which maps the elements of I to the corresponding generators of B_{W_{I}} and a_{b,s} to 1 since this is compatible with the relations in D_{I}. The homomorphism h is evidently surjective and its kernel contains U_{I}. On the other hand there exists also a homomorphism j : B_{W_{I}} → B_{I} which maps the generators of B_{W_{I}} to the elements of I. We have h ∘ j = Id and as j is clearly surjective we obtain (i). This also shows that B_{I} does not intersect the kernel of h. Since U_{I} is a normal subgroup included in the kernel of h and D_{I} is generated by B_{I} and U_{I}, we have (ii).

Corollary 2.7. Consider a sequence of subsets of S, ∅ = I_{0} ⊊ I_{1} ⊊ I_{2} … ⊊ I_{n} = S and let U_{j} be the group U_{I} of 2.6 relative to the group B_{I_{j}}, with I = I_{j−1}; we have P_{W} = U_{n} ⋊ (U_{n−1} ⋊ (⋯ ⋊ (U_{2} ⋊ U_{1}) …)).

Proof: As U_{n} ⊂ P_{W}, corollary 2.6 applied with I = I_{n−1} implies that P_{W} = U_{n}⋊P_{W_{I_{n−1}}} . By induction we obtain the result.

3. Application to the groups of finite type. We study more precisely the case when W is finite using 2.7 to obtain a better presentation of the pure braid group than the one in 2.5. At first we prove that when W is finite, U_{I} in 2.6 is generated by the a_{b,s} (without need to take the normal closure). This is not true in general. For example let W be of type Ã_{1} and S = {s, t} ⊃ I = {s}. Then B_{W} is the free group on the generators {s, t} and U_{I} is the normal subgroup generated by a_{<t,s>^{(i)},r} for i = 0, 1, …, where r is equal to the last term of < t, s >^{(i+1)}. Then the element st^{2}s^{−1} is in U_{I} but is not contained in the subgroup generated by the a_{<t,s>^{(i)},r}, since any element in this subgroup begins with t^{±1}.

We begin with some general lemmas in which we do not assume W finite.

Lemma 3.1. Let s′ ∈ I, b ∈ B_{W}^{red} and s ∈ S be such that bs ∈ B_{W}^{red} is I-reduced; we have a relation of type (2) in 2.4 in which s′ and a_{b,s} appear if and only if b^{−1}s′b ∈ B_{{s,t}} for some t ≠ s such that m_{s,t} is finite.

Proof: If we have a relation of type (2), then the element b can be written as b_{0}< s, t >^{(i)} where b_{0}s is I-reduced, b_{0}t = s′b_{0} and m_{s,t} is finite. Hence we have

b^{−1}s′b = (< s, t >^{(i)})^{−1}t< s, t >^{(i)} ∈ B_{{s,t}}.

Conversely if b^{−1}s′b ∈ B_{{s,t}}, then we have b = b_{0}< s, t >^{(i)} or b = b_{0}< t, s >^{(i)} for some reduced-{s, t} element b_{0}, so b_{0}^{−1}s′b_{0} ∈ B_{{s,t}}, that is, s′b_{0} = b_{0}x with x ∈ B_{{s,t}}.

We have s′b_{0} ∈ B_{W}^{red} since b, hence b_{0} is I-reduced. Taking the images in W we get s′b_{0} = b_{0}p(x), with evident notation. As l(s′b_{0}) = l(b_{0}) + 1 and l(b_{0}p(x)) = l(b_{0}) + l(p(x)) we have l(p(x)) = 1, so p(x) is equal to s or t. Write p(x) = r ∈ {s, t}. Using the section q we get s′b_{0} = b_{0}r, so x = r and, as bs is I-reduced, we have r = t if b = b_{0}< s, t >^{(i)} and r = s if b = b_{0}< t, s >^{(i)}. Hence the conjugation of the element a_{b,s} by s′ appears among the relations of type (2) in 2.4.

We remark that in the above lemma b^{−1}s′b cannot be equal to 1 or s, thus there is no ambiguity on t.

Lemma 3.2. Let t, t′ ∈ S and w, w′ ∈ B_{W}^{red} be such that wt, w′t′ ∈ B_{W}^{red} and p(wtw^{−1}) = p(w′t′w^{′−1}) where p is the homomorphism from B_{W} to W, then wtw^{−1} and w′t′w^{′−1} are conjugate by a product of elements of the form vs^{2}v^{−1} with s ∈ S, vs ∈ B_{W}^{red} and l(v) < l(w).

Proof: Write w = s_{1}s_{2} … s_{l} with s_{i} ∈ S. We will prove by induction on the length of w that wtw^{−1} and w′t′w^{′−1} are conjugate by a product of elements of the form s_{1} … s_{k−1}s_{k}^{2}s_{k−1}^{−1} … s_{1}^{−1} with 1 ≤ k ≤ l. If w is the identity element, by assumption we have p(tw′) = p(w′t′) and w′t′ ∈ B_{W}^{red}, which implies tw′ = w′t′, that is, t = w′t′w^{′−1}. Assume now that l ≥ 1. If s_{1}w′ ∉ B_{W}^{red}, we define w′′ ∈ B_{W}^{red} to be the element which satisfies w′ = s_{1}w′′, and if s_{1}w′ ∈ B_{W}^{red}, we put w′′ = s_{1}w′. We claim that w′′t′ ∈ B_{W}^{red}. This is trivial when s_{1}w′ ∉ B_{W}^{red} since s_{1}w′′t′ = w′t′ ∈ B_{W}^{red}. When w′′ = s_{1}w′ ∈ B_{W}^{red}, if w′′t′ = s_{1}w′t′ ∉ B_{W}^{red}, then we have s_{1}w′ = w′t′ by lemma 1.1 and the assumption p(wtw^{−1}) = p(w′t′w^{′−1}) = p(s_{1}) implies p(wt) = p(s_{1}w) which contradicts wt ∈ B_{W}^{red} and s_{1}w = s_{1}^{2}s_{2} … s_{k} ∉ B_{W}^{red}, whence our claim in this case. By induction hypothesis we get then that (s_{1}^{−1}w)t(s_{1}^{−1}w)^{−1} and w′′t′w^{′′−1} are conjugate by a product of elements of the form s_{2} … s_{k−1}s_{k}^{2}s_{k−1}^{−1} … s_{2}^{−1} with 2 ≤ k ≤ l. Applying the conjugation by s_{1}, we obtain the required result as s_{1}w′′ is either equal to w′ or to s_{1}^{2}w′.

Lemma 3.3. For I ⊂ S, let U′_{I} be the subgroup of D_{I} generated by the a_{b,s} and X_{I} the set of reflections p(bsb^{−1}) for s ∈ S and b ∈ B_{W}^{red} such that bs ∈ B_{W}^{red} is I-reduced. For each element w in X_{I} we choose s_{w} ∈ S and b_{w} ∈ B_{W}^{red} such that b_{w}s_{w} ∈ B_{W}^{red} and p(b_{w}s_{w}b_{w}^{−1}) = w. Then U′_{I} is already generated by the a_{b_{w},s_{w}} for w ∈ X_{I}.

Proof: We prove by induction on the length of b that each a_{b,s} such that bs ∈ B_{W}^{red} is Ireduced is contained in the subgroup generated by the a_{b_{w},s_{w}}. If b is the identity element, then we have p(s) = p(b_{s}s_{s}b_{s}^{−1}) whence p(sb_{s}) = p(b_{s}s_{s}), which implies sb_{s} = b_{s}s_{s}, i.e., s = b_{s}s_{s}b_{s}^{−1}. Now we assume that l(b) > 0 and put w = p(bsb^{−1}) ∈ X_{I}. By lemma 3.2 bsb^{−1} and b_{w}s_{w}b_{w}^{−1} are conjugate by a product of some a_{b′,s′} with l(b′) < l(b). By induction hypothesis we get the result.

Corollary 3.4. For each reflection w ∈ W, we choose s_{w} ∈ S and b_{w} ∈ B_{W}^{red} such that b_{w}s_{w} ∈ B_{W}^{red} and p(b_{w}s_{w}b_{w}^{−1}) = w. Then P_{W} is generated by the a_{b_{w},s_{w}}.

Proof: Apply lemma 3.3 with I = ∅.

Remark 3.5. When W is finite, the set X_{I} in 3.3 is exactly the set of reflections which are not contained in W_{I}. Let b^{I} = w_{I}^{−1}w_{S} ∈ B_{W}^{red} be the longest I-reduced element, and let us fix a reduced expression b^{I} = s_{1}s_{2} … s_{l}. Then the set X_{I} consists of the reflections s_{1}s_{2} … s_{i}s_{i−1} … s_{1} for 1 ≤ i ≤ l. In fact if bs is I-reduced, then there exists a reduced expression of b^{I} which begins with bs, thus the reflection p(bsb^{−1}) is in the predicted set. The converse is trivial. This result is not true for general Coxeter groups. For example in case of type Ã_{2}, let S = {r, s, t} and I = {r, s}. The reflection srtrs is not in W_{I} but it cannot be written as p(ws′w^{−1}) with a I-reduced element ws′. In fact if this was possible, we would have rsws′ = trsw ∈ B_{W}^{red} which is divisible by rs and t. Let then v ∈ B_{W}^{red} be an element divisible by rs and t and having shortest length with this property. Then there exists v′ ∈ B_{W}^{red} such that v = rtrv′ which implies that trv′ is divisible by s and tr. By symmetry of s, t and r, this contradicts the choice of v.

Proposition 3.6. If W is finite, then the group U_{I} is generated by the a_{b,s}.

Proof: Let s′ be an element of I. It is enough to prove that s′ normalizes the subgroup U′_{I} in 3.3. Let b^{I} = w_{I}^{−1}w_{S} ∈ B_{W}^{red} be the longest I-reduced element and s_{1} an element of S which divides b^{I} in B_{W}^{+} . Then we have s′b^{I} = b^{I}r for some r ∈ S, since w_{I} (resp. w_{S}) conjugates I (resp. S) on itself. Thus s′b^{I} is divisible on the left by s′ and s_{1}, hence by w_{{s′,s_{1}}}. Therefore we have b^{I} =< s_{1}, s′; m_{s_{1},s′} − 1 > b_{1} and dividing by < s_{1}, s′; m_{s_{1},s′} − 1 >, the equation s′b^{I} = b^{I}r implies s′_{2}b_{1} = b_{1}r ∈ B_{W}^{red}, where s′_{2} is equal to the element of the set {s′, s_{1}} which satisfies s′w_{{s_{1},s′}} = w_{{s_{1},s′}}s′_{2}. Iterating the same argument, we can write b^{I} = ∏_{i=1}^{i=k} < s_{i}, s′_{i}; m_{s_{i},s′_{i}} − 1 > where s′_{1} = s′ and s′_{i+1} is equal to the element of {s′_{i}, s_{i}} which satisfies s′_{i}w_{{s_{i},s′_{i}}} = w_{{s_{i},s′_{i}}}s′_{i+1}. We note that the element ∏_{i=1}^{i=h−1} < s_{i}, s′_{i}; m_{s_{i},s′_{i}} − 1 > is reduced-{s_{h}, s′_{h}} for 1 ≤ h ≤ k. Remark 3.5 says that the set X_{I} in 3.3 consists of the reflections p(bsb^{−1}) where bs = [∏_{i=1}^{i=h−1} < s_{i}, s′_{i}; m_{s_{i},s′_{i}} − 1 >] < s_{h}, s′_{h}; j >, 1 ≤ h ≤ k and 1 ≤ j ≤ m_{s_{h},s′_{h}} − 1 and s is equal to the last term of < s_{h}, s′_{h}; j >. This implies that

s′b = [∏_{i=1}^{i=h−1} < s_{i}, s′_{i}; m_{s_{i},s′_{i}} − 1 >] s′_{h} < s_{h}, s′_{h}; j − 1 >, hence the element b^{−1}s′b = (< s_{h}, s′_{h}; j − 1 >)^{−1}s′_{h} < s_{h}, s′_{h}; j − 1 > is in a group of the form B_{{s,t}} with finite m_{s,t}. Then we can apply lemma 3.1: by the relations of type (2) in 2.4 the conjugate of a_{b,s} by s′ is a product of the elements of the same form. Hence s′ normalizes U′_{I} which proves the proposition.

Remark. If the element b^{I} has a unique expression as a product of generators in B_{W}^{+} , then no relation of type (1) in 2.4 can exist in D_{I}. The cases in which this condition is satisfied are exactly the cases A_{n}, B_{n} and I_{2}(m) with I = {s_{1}, s_{2}, …, s_{n−1}} (the double edge is between s_{1} and s_{2} for B_{n}).

By a case by case argument we shall now give a presentation of D_{I} when W is of type A_{n}, B_{n}, D_{n} or I_{2}(m) and W_{I} is a maximal parabolic subgroup of the same type as W.

We first review the case A_{n} which is well-known.

Type A_{n}. The Coxeter diagram is

•^{s_{1}} ⋯ •^{s_{n−1}} •^{s_{n}}.

We take I = {s_{1}, …, s_{n−1}}. Then b^{I} = s_{n}s_{n−1} … s_{1}. The a_{b,s} are the a_{i} = (^{s_{n}...s_{i+1}}s_{i})^{2} for i = 1, …, n. Then the relations in D_{I} are the braid relations in I and the following, where 1 ≤ i ≤ n − 1 and 1 ≤ j ≤ n

s_{i}a_{j}s_{i}^{−1} = a_{j}  if j ≠ i, i + 1
s_{i}a_{i}s_{i}^{−1} = a_{i+1}
s_{i}a_{i}a_{i+1}s_{i}^{−1} = a_{i}a_{i+1}

Type B_{n}. The Coxeter diagram is

•^{s_{1}} •^{s_{2}} ⋯ •^{s_{n}}.

We take I = {s_{1}, …, s_{n−1}}. Then b^{I} = s_{n}s_{n−1} … s_{1}s_{2} … s_{n}. The a_{b,s} are the a_{i} = (^{s_{n}...s_{i+1}}s_{i})^{2} for i = 1, …, n and the b_{i} = (^{s_{n}...s_{1}s_{2}...s_{i−1}}s_{i})^{2} for i = 2, …, n. Then the relations in D_{I} are the braid relations in I and the following, where 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n and 2 ≤ h ≤ n

s_{i}a_{j}s_{i}^{−1} = a_{j}  if j ≠ i, i + 1
s_{i}a_{i}s_{i}^{−1} = a_{i+1}  for i ≠ 1
s_{i}a_{i}a_{i+1}s_{i}^{−1} = a_{i}a_{i+1} for i ≠ 1
s_{i}b_{h}s_{i}^{−1} = b_{h}  if h ≠ i, i + 1
s_{i}b_{i+1}s_{i}^{−1} = b_{i}  for i ≠ 1
s_{i}b_{i+1}b_{i}s_{i}^{−1} = b_{i+1}b_{i} for i ≠ 1
s_{1}b_{2}s_{1}^{−1} = a_{2}
s_{1}b_{2}a_{1}s_{1}^{−1} = a_{1}a_{2}
s_{1}b_{2}a_{1}a_{2}s_{1}^{−1} = b_{2}a_{1}a_{2}

Type I_{2}(m). We denote by s and t the two generators and we take I = {s}. We have b^{I} = < t, s >^{(m−1)} and the a_{b,s} are the a_{i} = (^{<t,s>^{(i−1)}}r)^{2}, for i = 1, …, m − 1 where r equals s if i is even and t otherwise. Proposition 2.4 gives relations of type (2) with b_{0} = 1 and s′ = s. Hence we obtain the relations

sa_{m−1}a_{m−2} … a_{m−i}s^{−1} = a_{i} … a_{2}a_{1}, for i = 1, …, m − 1.

Type D_{n}. The Coxeter diagram is

•^{s_{2}} _{•_{s_{2′}}} •^{s_{3}} •^{s_{4}} ⋯ •^{s_{n}}.

We suppose n ≥ 2. We denote the generators by s_{2}, s_{2′}, s_{3}, …, s_{n} and we take I = S−{s_{n}} if n ≥ 3 and I = ∅ if n = 2. We have b^{I} = s_{n}s_{n−1} … s_{3}s_{2}s_{2′}s_{3} … s_{n}. The element b^{I} has exactly two expressions in B_{W}^{+} , the other expression is obtained from the above one by exchanging s_{2} and s_{2′}. Hence the elements a_{b,s} are the a_{i} = (^{s_{n}...s_{i+1}}s_{i})^{2} for i = 2, 2′, 3, …, n, with the convention that i+1 = 3 if i = 2′ and the b_{i} = (^{s_{n}...s_{3}s_{2}s_{2′} s_{3}...s_{i−1}}s_{i})^{2} for i = 3, …, n. We note that the two elements a_{s_{n}...s_{3}s_{2′} ,s_{2}} and a_{s_{n}...s_{3}s_{2},s_{2′}} are equal respectively to a_{2} and a_{2′}.

We apply proposition 2.4. For relations of type (1), the only possible triples (b_{0}, s, t) are (s_{n} … s_{4}s_{3}, s_{2}, s_{2′}) and (s_{n} … s_{4}s_{3}, s_{2′}, s_{2}), which gives the relation a_{2}a_{2′} = a_{2′}a_{2}.

The relations of type (2) are as follows, where i ranges over {2, 2′, 3, …, n − 1} and j ranges over {2, 2′, 3, …, n} (we also make the convention that i + 1 = 3 if i = 2′):

s_{i}a_{j}s_{i}^{−1} = a_{j}  if j ≠ i, i + 1 and {i, j} ≠ {2, 2′}
s_{i}a_{i}s_{i}^{−1} = a_{i+1}
s_{i}a_{i}a_{i+1}s_{i}^{−1} = a_{i}a_{i+1}
s_{i}b_{j}s_{i}^{−1} = b_{j}  if j ≠ i, i + 1 and j ≥ 3
s_{i}b_{i+1}s_{i}^{−1} = b_{i}  for i ≥ 3
s_{i}b_{i+1}b_{i}s_{i}^{−1} = b_{i+1}b_{i} for i ≥ 3
s_{2}b_{3}s_{2}^{−1} = a_{2′}
s_{2}b_{3}a_{2′}s_{2}^{−1} = b_{3}a_{2′}

as well as the relations obtained from the two last lines by exchanging a_{2} and a_{2′} and at the same time s_{2} and s_{2′}.

Hence the group D_{I} has a presentation where the generators are I, the a_{i} and the b_{i} and the relations are the braid relations in I, the relation a_{2}a_{2′} = a_{2′}a_{2} and the relations above.

We now give more information on U_{I} for types A, B, I_{2}.

Proposition 3.7. If the type of (W, S) is A_{n}, B_{n} or I_{2}(m) and I is as above, the group U_{I} is the free group generated by the a_{b,s} where bs is I-reduced.

Proof: For type A_{n} this is a known result and for type I_{2}(m) it is immediate from the presentation given above. We study the case B_{n}. The method which we use can also be applied to A_{n}. We start by changing the set of generators, putting x_{i} = b_{n}b_{n−1} … b_{2}a_{1}a_{2} … a_{i} for i = 1, …, n and y_{i} = b_{n}b_{n−1} … b_{i+1} for i = 1, …, n − 1. The group U_{I} is generated by the x_{i} and the y_{i} and D_{I} has a presentation with generators the x_{i} (with i = 1, …, n), the y_{i} (with i = 1, …, n − 1) and the s_{i} (with i = 1, …, n − 1) and with relations the braid relations between the s_{i} and the relations of type (2) which are

s_{j}x_{i}s_{j}^{−1} = x_{i}  if i ≠ j
s_{j}y_{i}s_{j}^{−1} = y_{i}  if i ≠ j
s_{i}x_{i}s_{i}^{−1} = x_{i−1}x_{i}^{−1}x_{i+1} for i ≠ 1
s_{i}y_{i}s_{i}^{−1} = y_{i+1}y_{i}^{−1}y_{i−1} for i ≠ 1
s_{1}x_{1}s_{1}^{−1} = y_{2}y_{1}^{−1}x_{2}
s_{1}y_{1}s_{1}^{−1} = y_{2}x_{1}^{−1}x_{2}

In order to show that U_{I} is the free group generated by the a_{i} and the b_{i} or, by the x_{i} and the y_{i}, it is sufficient to show that the formulas above can be used to define an action of B_{I} on a free group. More precisely it is sufficient to show that if F is the free group generated by {ξ_{1}, …, ξ_{n}, η_{1}, …, η_{n−1}}, then the formulas above with x_{i} replaced by ξ_{i} and y_{i} replaced by η_{i} define an action of B_{I} on F. The group F ⋊B_{I} has then the same presentation as D_{I} hence they are isomorphic, the isomorphism mapping F onto U_{I}.

The action of a fixed s_{j} given above is an automorphism of F. Thus it is sufficient to show that these automorphisms satisfy the braid relations. The commutation of the actions of s_{j} and s_{k} when |j − k| > 1 is clear. To check the remaining braid relations we use the next lemma which results from a simple calculation.

Lemma 3.8. Let s and t be two automorphisms of a free group G generated by four elements w, x, y and z. We suppose that

(i) s acts trivially on x, w and z and maps y to xy^{−1}z. (ii) t acts trivially on y, w and z and maps x to wx^{−1}y

Then s ∘ t ∘ s = t ∘ s ∘ t.

Applying the lemma with s = s_{j} and t = s_{j+1}, acting on the free group generated by w = η_{j+2} (resp. w = ξ_{j−1}), x = η_{j+1} (resp. x = ξ_{j}), y = η_{j} (resp. y = ξ_{j+1}) and z = η_{j−1} (resp. z = ξ_{j+2}) with j ≥ 2, we obtain that the actions of s_{j} and of s_{j+1} satisfy the braid relation. It remains to check the braid relation between s_{1} and s_{2}. We can verify that the composition of automorphisms s_{1} ∘ s_{2} ∘ s_{1} ∘ s_{2} and s_{2} ∘ s_{1} ∘ s_{2} ∘ s_{1} map ξ_{1} to η_{3}η_{1}^{−1}ξ_{3} (resp. ξ_{2} to η_{3}η_{2}^{−1}ξ_{3}, resp. η_{1} to η_{3}ξ_{1}^{−1}ξ_{3}, resp. η_{2} to η_{3}ξ_{2}^{−1}ξ_{3}).

Remark. In the D_{n} case with I as above, the group U_{I} is evidently not free because of the relation a_{2}a_{2′} = a_{2′}a_{2}, and yet it is not the group generated by the a_{b,s} with this only relation. For example the elements a_{2′}^{−1}b_{3}a_{2′} and a_{3} also commute (they are the images of a_{2′} and a_{2} by the conjugation by s_{2}). A similar remark can be made in all cases except those of proposition 3.7.

We finish this section with a property of the pure braid groups of type A_{n}, B_{n} and D_{n}. If we take I as above, the group B_{I} is a group of same type, of rank n − 1 (except the case D_{2}), so we can consider an increasing sequence of subsets of S, I_{0} = ∅ ⊂ I_{1} … ⊂ I_{n−1} ⊂ I_{n} = S where |I_{i}| = i except the case i = 1 for the type D_{n} with I_{1} = I_{0} = ∅. We denote by U_{j} as in 2.7 the group U_{I} when the ambient group is B_{I_{j}} and I = I_{j−1}.

Proposition 3.9. With notation as above, for i ≥ 2, the conjugation by s_{i} is an isomorphism of U_{i−1} on a subgroup of U_{i} which is compatible with the action of B_{I_{i−2}}.

Proof: By the formulas for the generators of U_{i−1}, the conjugation maps U_{i−1} in U_{i}. As s_{i} centralizes B_{I_{i−2}}, we obtain the result.

Using proposition 3.7 and the fact that, with notation as above we have P_{W} = U_{n} ⋊ (U_{n−1}⋊(⋯⋊(U_{2}⋊U_{1}) …)), we can obtain a presentation of P_{W} when W is of type A_{n}, B_{n} and I_{2}(m), using the standard method for getting a presentation of a semi-direct product (cf. e.g., [H, appendix by Gædde, 4]). The method says that P_{W} has a presentation where the set of generators is the union of the sets of generators of the groups U_{i} for i = 1, 2, …, n and the only relations are those giving the action by conjugation of the generators of U_{i} on the generators of U_{j} for 1 ≤ i < j ≤ n. We give below the results for types B_{n} and I_{2}(m). Type A_{n} can be seen inside type B_{n+1}.

Presentation of the pure braid group of type B_{n}. We get a presentation of the pure braid group of type B_{n} as an n-fold semi-direct product of free groups. The set of generators consists of the a_{i,j} with 1 ≤ i ≤ j ≤ n and of the b_{i,j} with 2 ≤ i ≤ j ≤ n, where a_{i,j} = ^{s_{i}s_{i−1}...s_{j+1}}s_{j}^{2} and b_{i,j} = ^{s_{i}s_{i−1}...s_{2}s_{1}s_{2}...s_{j−1}}s_{j}^{2}. The group generated by the a_{i,j} and b_{i,j} with fixed index j is normalized by all the a_{h,k} and b_{h,k} with k < j; it is the subgroup U_{I} of the braid group of type B_{j} when I = {s_{1}, … s_{j−1}}. We get all the relations in the pure braid group if we know the actions by conjugation of a_{h,k} and b_{h,k} on a_{i,j} and b_{i,j} for all indices such that k < j. We give it in the following tables. The computation is tedious but easy from the formulas for the action of the elements s_{l}. In the tables we fix j and put a_{i} = a_{i,j} and b_{i} = b_{i,j}. The image of a_{i} (resp. b_{i}) by conjugation by a_{h,k} is always a conjugate of a_{i} (resp. b_{i}) in the group U_{I}. So in the tables we give for each a_{h,k} and each b_{h,k} the element of U_{I} which has same action on a_{i} (resp. b_{i}) by left conjugation. Conjugation of a_{i} by a_{h,k}: it is equal to the conjugation of a_{i} by

• 1 if i ≤ h − 1 or i ≥ k + 2.

• (a_{h}a_{k+1})^{−1} if i = k + 1, h ≠ 1.

• a_{k+1}^{−1}a_{1}^{−1}(b_{k+1}b_{k} … b_{2})^{−1}b_{k} … b_{3}b_{2} if i = k + 1, h = 1.

• a_{k+1}^{−1}a_{h}^{−1}a_{k+1}a_{h} if h + 1 ≤ i ≤ k, h ≠ 1.

• a_{k+1}^{−1}a_{1}^{−1}(b_{k+1}b_{k} … b_{2})^{−1}b_{k} … b_{3}b_{2}a_{k+1}(b_{k} … b_{3}b_{2})^{−1}b_{k+1}b_{k} … b_{2}a_{1} if 2 ≤ i ≤ k, h = 1.

• a_{k+1}^{−1} if i = h ≠ 1.

• a_{k+1}^{−1}a_{1}^{−1}(b_{k+1} … b_{3}b_{2})^{−1}b_{k} … b_{3}b_{2} if i = h = 1. Conjugation of a_{i} by b_{h,k}: it is equal to the conjugation of a_{i} by

• 1 if i ≥ k + 2.

• a_{k+1}^{−1}b_{h}^{−1} if i = k + 1.

• a_{k+1}^{−1}b_{h}^{−1}a_{k+1}b_{h} if 1 ≤ i ≤ k and i ≠ h.

• a_{k+1}^{−1}b_{h}^{−1}a_{k+1}b_{h}(b_{k+1}b_{k} … b_{2}a_{1}a_{2} … a_{h})^{−1}b_{k} … b_{3}b_{2}a_{1}a_{2} … a_{h−1} if i = h. Conjugation of b_{i} by a_{h,k}: it is equal to the conjugation of b_{i} by

• 1 if i ≥ k + 2 or i ≤ h − 1 or h + 1 ≤ i ≤ k.

• (b_{k+1} … b_{h+1}b_{h})^{−1}b_{k} … b_{h+1}b_{h} if i = h.

• b_{k}b_{k−1} … b_{h+1}(b_{k} … b_{h+1}b_{h})^{−1} if i = k + 1 and h ≠ 1.

• b_{k}b_{k−1} … b_{2}a_{k+1}^{−1}a_{1}^{−1}(b_{k}b_{k−1} … b_{2})^{−1} if i = k + 1 and h = 1. Conjugation of b_{i} by b_{h,k}: it is equal to the conjugation of b_{i} by

• 1 if i ≥ k + 2 or h + 1 ≤ i ≤ k.

• a_{k+1}^{−1}b_{h}^{−1}a_{k+1}b_{h} if i ≤ h − 1.

• a_{k+1}^{−1} if i = h.

• b_{k} … b_{3}b_{2}a_{1}a_{2} … a_{h}(b_{k} … b_{3}b_{2}a_{1}a_{2} … a_{h+1})^{−1} if i = k + 1.

Remark. The elements a_{i,j} with 1 ≤ i ≤ j above generate the pure braid group of type A_{n−1} and the presentation of this group can be extracted form the above formulas by keeping only the cases where h ≠ 1 in the first paragraph.

Presentation of the pure braid group of type I_{2}(m). This group is the semi-direct product of a free group generated by a_{1}, … a_{m−1} by a free group generated by an element b, where a_{i} = (^{<t,s>^{(i−1)}}r)^{2}, with r equal to s if i is even and t otherwise, and where b = s^{2}.

The presentation consists of this set of generators and the relations expressing the action of b by conjugation on a_{i} for each i, which is given by the following proposition.

Proposition 3.10. The element b acts on the free group generated by a_{1}, …, a_{m−1} by conjugation by the product (a_{m−1} … a_{2}a_{1})^{−1}.

Proof: We know that the element s conjugates a_{m−1}a_{m−2} … a_{m−i} on a_{i} … a_{2}a_{1} for any i. The latter element can be written (a_{m−1} … a_{i+1})^{−1}a_{m−1} … a_{2}a_{1}. This element is sent by another conjugation by s on (a_{m−i−1} … a_{1})^{−1}a_{m−1} … a_{1} which can be written (a_{m−i−1}...a_{1})^{−1}(a_{m−1} … a_{m−i}) = ^{(a_{m−1}...a_{1})^{−1}}(a_{m−1} … a_{m−i}). As the free group is generated also by the elements a_{m−1}a_{m−2} … a_{m−i} for i = 1, … m − 1, we get that s^{2} acts on the free group as stated in the proposition.

4. Embedding the group of type B_{n} into the group of type A_{n}.

In this section we show that the actions of braid groups of type A or B on a free group defined above are faithful and we recover in this way the embedding of the braid group of type B_{n} into the braid group of type A_{n}. More precisely, we note that when W is of type A_{n} and I = {s_{1}, s_{2}, …, s_{n−1}}, the subgroup D_{I} is isomorphic to the braid group of type B_{n} with generators {s_{1}, …, s_{n−1}, s_{n}^{2}}. In fact the presentation of D_{I} given in section 2 shows that D_{I} is generated by {s_{1}, …, s_{n−1}, s_{n}^{2}} and when we rewrite the relations using these generators, we can obtain exactly the relations of a braid group of type B_{n}. We want to show a compatibility of this embedding with the actions of the two braid groups on free groups.

Proposition 4.1. The actions of the braid groups of type A or B on the free groups defined above are faithful.

Proof: This result is known for the type A. We prove it for the type B. The proof for the type A is analogue. First of all we note that the action of the braid group of type B_{n−1} on the free group with 2n − 1 generators defined above induces the action of the Coxeter group of type B_{n−1} on the free abelian group of rank 2n − 2 whose generators are the images of a_{2}, …, a_{n} and b_{2}, …, b_{n}. This action is the natural action of the Coxeter group of type B_{n−1} by permutation of 2n−2 elements. It is faithful. Therefore an element of the braid group which acts trivially on the free group must be in the pure braid group P_{n−1} of type B_{n−1}. We decompose the pure braid group P_{n} as a semi-direct product U_{n} ⋊ P_{n−1}, where U_{n} is the free group generated by a_{1}, …, a_{n} and b_{2}, …, b_{n}. Thus to find the kernel of the action is equivalent to find the elements of P_{n−1} which commute with U_{n} in P_{n}.

Let b ≠ 1 be in P_{n−1} acting trivially on U_{n} and let i be the smallest index such that b is in P_{i}. Using the conjugation by s_{n}s_{n−1} … s_{i+2} and applying 3.9 we see that b commutes with U_{i+1}, thus we can reduce the proof to the case when i = n − 1.

The element b can be written uniquely as b_{n−1}b_{n−2} … b_{0} with b_{j} ∈ U_{j} for all j and b_{n−1} ≠ 1. As b commutes with U_{n}, it commutes in particular with a_{n}. Since all elements of B_{I_{j}} commute with a_{n} = s_{n}^{2} for j = 1, …, n − 2, we see that b commutes with a_{n} if and only if b_{n−1} commutes with a_{n}. Conjugating by s_{n}, we obtain an element ^{s_{n}}b_{n−1} ∈ U_{n} which commutes with a_{n}. Since the group U_{n} is a free group in which a_{n} is one of the generators, ^{s_{n}}b_{n−1} must be a power of a_{n} = s_{n}^{2}. This implies that b_{n−1} is a power of a_{n}, which contradicts b_{n−1} ∉ U_{n}.

let us consider now the braid group of type A_{n} with generators s_{1}, …, s_{n} and the braid group of type B_{n} with generators s′_{1}, …, s′_{n}. We have s_{1}^{2}s_{2}s_{1}^{2}s_{2} = s_{2}s_{1}^{2}s_{2}s_{1}^{2}, thus there exists a homomorphism φ from the braid group of type B_{n} to the braid group of type A_{n} which maps s′_{1} to s_{1}^{2} and s′_{i} to s_{i} for i ≥ 2. Our aim is to show the next result.

Theorem 4.2. The homomorphism φ is injective.

The theorem follows from the next proposition.

Proposition 4.3. let F be a free group on n + 1 generators a_{1}, …, a_{n+1}, provided with the action of the braid group of type A_{n} defined above and let F′ be a free group on 2n+1 generators a′_{1}, … a′_{n+1}, b_{2} … b_{n+1} provided with the action of the braid group of type B_{n} as above; then

(i) the homomorphism ψ of F′ in F defined by

\{ \begin{matrix}ψ(a′_{1}) = a_{1}^{2}, \\ ψ(a′_{i}) = a_{i}, & for i = 2 … n + 1 \\ ψ(b_{i}) = a_{1}a_{2} … a_{i}a_{i−1}^{−1} … a_{2}^{−1}a_{1}^{−1} & for i = 2 … n + 1 \end{matrix}

is injective. (ii) the image ψ(F′) is a normal subgroup in F of index 2. (iii) the action of the braid group of type B_{n} on F through φ stabilizes ψ(F′) and ψ is

compatible with the action of the braid group of type B_{n}.

The theorem is immediate from the proposition since by (iii) and (i) an element of ker φ acts trivially on F′ and the action on F′ is faithful by 4.1. Proof of 4.3: Assertion (iii) results from a simple calculation. In order to prove (i) and (ii) we change the generators in F. We put x_{i} = a_{1} … a_{i} for i = 1, 2, …, n + 1. The group F is also the free group generated by x_{1}, …, x_{n+1}. We have

\{ \begin{matrix}ψ(a′_{1}) = x_{1}^{2}, \\ ψ(a′_{i}) = x_{i−1}^{−1}x_{i}, & for i = 2 … n + 1 \\ ψ(b_{i}) = x_{i}x_{i−1}^{−1} & for i = 2 … n + 1. \end{matrix}

Therefore ψ(F′) contains all the products x_{i}x_{j}, x_{i}^{−1}x_{j} and x_{i}x_{j}^{−1} with i, j ∈ {1, …, n+1}. It is the subgroup of F consisting of the elements of even length with respect to the x_{i} and their inverses. This subgroup is of index 2 in F. In order to prove (i) we apply the Reidemeister-Schreier method to obtain a presentation of ψ(F′). The set {1, x_{1}} forms a set of representatives of the quotient ψ(F′)\F. Then we obtain the system of generators by multiplying x_{i} and x_{1}x_{i} by the inverses of their representatives which are respectively x_{1} and 1. Thus we obtain x_{i}x_{1}^{−1} for i = 2, …, n + 1 and x_{1}x_{i} for i = 1, …, n + 1. The method says that there are no relations between these generators. Hence ψ(F′) is the free group generated by these elements, hence also the free group generated by x_{1}^{2}, the x_{i−1}^{−1}x_{i} and the x_{i}x_{i−1}^{−1} for i = 2 … n + 1, since this last set is a system of generators and has the right cardinality.

As ψ(F′) is the free group generated by the (distinct) images of the generators of F′, the homomorphism ψ is injective.

Remark. One can also prove this last proposition topologically by interpreting the braid group of type B_{n} as a subset of braids with n + 1 strings. We wanted to give a purely combinatorial proof.

References.

[Bi] J. S. Birman, Braids links and mapping class groups, Annals of Mathematics Studies 82 (1975).

[BrMaRo] M. Broué, G. Malle and R. Rouquier, Complex reflection groups, braid groups, Hecke algebras J. Reine Angew. Math. 500 (1998) 127–190.

[H] V.L. Hansen, Braids and coverings, selected topics, London Math. Society Student Texts 88, Cambridge University Press (1989).

[J] D.L. Johnson, Presentation of Groups, London Math. Society Student Texts 15, Cambridge University Press (1990).

LAMFA, Faculté de mathématiques et informatique Université de Picardie-Jules Verne 33 Rue Saint Leu 80039 Amiens France

digne@u-picardie.fr

Department of Mathematics Faculty of science and technology Sophia University 7-1 Kioicho, Chiyoda Ku Tokyo, 102-8554 Japan

gomi@mm.sophia.ac.jp