A Product on Double Cosets of $B_\infty$

For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\backslash G/K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_\infty$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.


The infinite braid group and double cosets
The Artin braid group on n strings B n has the presentation with n − 1 generators σ 1 , σ 2 , . . . , σ n−1 and the so-called braid relations: σ i σ j = σ j σ i , |i − j| ≥ 2, i, j ∈ {1, . . . , n − 1}, The generators σ i are called elementary braids. For each n, consider the monomorphism i n : B n → B n+1 sending the k-th elementary braid of B n to the k-th elementary braid of B n+1 . Geometrically this operation corresponds to adding a new string to the right of the braid, without creating any new crossings, as in the picture below: . . . The direct limit of this sequence of groups, with respect to the homomorphisms i n , is the infinite braid group consisting of braids with countably many strings and finitely many crossings. This group has the presentation: For each non-negative integer α, let B ∞ [α] be the subgroup of B ∞ given by Definition 1.1. Let G be a group, g ∈ G and K and L be subgroups of G. The double coset on G containing g with respect to the pair (K, L) is the set KgL. Denote by K\G/L the set of double cosets on G with respect to the pair (K, L).

The Burau representation of B ∞
The Burau representation is the homomorphism η n : B n → GL(n, Z[t, t −1 ]) given by Consider the homomorphisms j n : GL(n) → GL(n + 1) given by j n (T ) = T 0 0 1 .
The group GL(∞) is the direct limit of GL(n) with respect to the homomorphisms j n and consists of infinite matrices that differ from the identity matrix only in finitely many entries. Due the commutativity of the diagram: / / GL(n + 1) we can construct a representation η : B ∞ → GL(∞) of B ∞ by taking the limit of the representations η n . More precisely, η is given by the following formulas: With this representation in mind we will define an operation on the double cosets of GL(∞) such that the Burau representation will be functorial between the categories of double cosets.

Main results
Further we define the element θ n [β] ∈ B ∞ [β] as: Finally, the definition of the product of the double cosets is as follows: be double cosets. Consider p ∈ p and q ∈ q representatives of these double cosets. Then we define their product as for sufficiently large n.
Theorem 1.4. The operation defined above does not depend on the choice of the representatives of the double cosets for n large enough. Moreover it is associative.
Remark 1.5. We will show that exists some n 0 (α, γ, p, q) such that for all n ≥ n 0 . We can make n 0 more precise. In fact n 0 = max{supp p, supp q, α, γ} + 1, where supp is the support of a braid, defined in 2.1.
There is a natural one-to-one correspondence between the set and the set of ∞ is the image of the subgroup B ∞ [α] by the diagonal map). Therefore the conjugacy problem in B ∞ is equivalent to the word problem in As a consequence of the existence of a solution for the conjugacy problem for the braid groups and the fact that the injections i n do not merge conjugacy classes (see González-Meneses [3]) we have, Proposition 1.6. The conjugacy problem for B ∞ has a solution.
Combining the observations above, we see that the word problem for Consider the subgroup of GL(∞) given by Then we define their product as: for sufficiently large j.
Theorem 1.8. The operation defined above does not depend on the choice of representatives of double cosets for j large enough. Moreover, it is associative.
Remark 1.9. We will show that exists some j 0 (n, m, p, q) such that G for all j ≥ j 0 . We can make j 0 more precise. In fact let N ∈ N be such that p and q can be written as diagonal block As a special case, when G is the bisymmetric group (the group that consists of pairs (g, h) of permutations of N such that gh −1 is a finite permutation) and K is its diagonal subgroup, we get a special category, called the train category of the pair (G, K). This category admits a transparent combinatorial description and encode information about the representations of the bisymmetric group (see [16], [10]).

Comments
do not always coincide for all choices of p ∈ p and q ∈ q. For instance σ 2 and σ 3 σ 2 are representatives of the same double coset in B ∞ [2]\B ∞ /B ∞ [2]. But σ 2 2 and σ 3 σ 2 σ 3 σ 2 represent distinct cosets. To see this we consider the permutation associated to each braid. For the braid σ 2 2 it is the identity and for the braid σ 3 σ 2 σ 3 σ 2 it is (432). Since no braid in B ∞ [2] permutes the point 2 we see that these are in fact distinct double cosets. Before we proceed we introduce the notion of of support which will be needed later.
Notice that p has no factors in B ∞ [supp p], hence p commutes with every element of B ∞ [1 + supp p]. Also, we can identify p with an element of B 1+supp p . We define supp 1 = 0.
Consider p and q representatives of the double cosets Proposition 2.2. The sequence (r j ) j≥1 defined above is eventually constant.
Proof. We are going to give a proof in several steps: Step 1. Given m > 0 we have τ In fact we have the equality: Step Step 4. Let M = max{supp p, supp q, α, γ} + 1. Suppose that, for some m ≥ M , we have r m = r M . We are going to show that r m = r m+1 . Let u and ℓ be like in step 3.
and they commute with p and q. Therefore: Hence r m = r m+1 . What proves Proposition 2.2.
The following technical lemma will be used in Lemma 2.4, which in turn is used in Proposition 2.5 and more extensively in Theorem 2.6.
be a family of sequences of positive integers such that υ i j < υ n k whenever i = n and k < j or i < n and k = j; in other words, the sequences (υ i j ) j≥1 are decreasing and the sequences (υ i j ) i≥1 are increasing. If µ j = ℓ k=1 σ υ j k and λ i = g k=1 σ υ k i , then µ 1 · · · µ g = P = λ 1 · · · λ ℓ . Proof. We prove the lemma by induction on the pair (g, ℓ). The statement is trivial for g = ℓ = 1. Assume it is true for (g, ℓ), we prove it is true for (g + 1, ℓ) and (g, ℓ + 1).
(i) For (g + 1, ℓ), notice that: x r for r > t, this fallows from the inequalities υ s r < υ g+1 r < υ g+1 t for s < g + 1. Therefore: (ii) For (g, ℓ + 1) we have: If y s = ℓ r=1 σ υ s r notice that y s σ υ t ℓ+1 = σ υ t ℓ+1 y s for s > t, this follows from the inequalities υ t ℓ+1 < υ s ℓ+1 < υ s r for r < ℓ + 1. Therefore: It will be useful to write the product P from Lemma 2.3 as a matrix, where the indices increase from right to left and from top to bottom.
Consider, for each positive integer m, the homomorphism C m : B ∞ → B ∞ given by C m (σ j ) = σ m+j . Then we have the following lemma.
i−1 we see that θ j [β] = λ 1 · · · λ j . As we saw in item (i), we have that What completes the proof.
Our next step is to prove that the product does not depend on the chosen representatives.
Proposition 2.5. Let p ′ and q ′ be other two representatives of p and q respectively. Consider the sequence Then there exists an integer N > 0 such that Proof. Since p and p ′ are representatives of the same double coset, there exist r ∈ B ∞ [α] and h ∈ B ∞ [β] such that p ′ = rph. In a similar way, there exist k ∈ B ∞ [β] and s ∈ B ∞ [γ] such that q ′ = kqs. Therefore, . Furthermore,h commutes with q and k, andk commutes with p and h. Now: Therefore, for all pairs (p, p ∈ p, q ∈ q and j sufficiently large. Finally, we are going to prove the associativity of the operation •.
Proposition 2.6. The product of double cosets is associative.
Throughout the rest of the proof we will use the symbol a ≡ b to signify that a and b are representatives of the same double coset of B ∞ [α]\B ∞ /B ∞ [γ], that is, we can find elements h ∈ B ∞ [α] and k ∈ B ∞ [γ] such that hak = b.
Using the notation of Lemma 2.3 we can write Using the same lemma, we can see that Notice also that P 4 = R 5 W , where Since supp P = 2k and R 5 ∈ B ∞ [2k + 1], R 5 P = P R 5 and we have aP bR 5 W c = aP R 5 bW c = R 5 aP bW c ≡ aP bW c.
Our next objective is to find elements E, A ∈ B ∞ such that aP 2 bP 3 c ≡ aP bEAW c.
Step 1. aP 2 bP 3 c = aP bELP 3 c. Consider the element and notice that P 2 = P F . Since F ∈ B ∞ [k] we see that bF = F b.
Step 2. LP 3 c ≡ CP 3 c for some C. In fact, consider Then L = CD and, since D ∈ B ∞ [2k + 1] and supp P 3 = 2k, we have DP 3 c = P 3 cD ≡ P 3 c. Hence LP 3 c ≡ CP 3 c.

Proof of Proposition 1.6
The conjugacy problem for the braid group has a solution (see [1], [2]). This fact yields a solution for the conjugacy problem in B ∞ . In fact, consider the monomorphisms I n : B n → B ∞ given by the direct limit (these homomorphisms consist of adding countably many strands to the right of the braid, without creating any new crossing). Then, for two braids p and q in B ∞ , there exists n ∈ N such that p = I n (x) and q = I n (y) for some x, y ∈ B n . If x and y are conjugate, there exists z ∈ B n such that x = zyz −1 . Hence, p = I n (z)yI n (z) −1 , that is, p and q are conjugated. Now, suppose that p and q are conjugated. Then p = rqr −1 for some r ∈ B ∞ . As before, there exists m ∈ N, with m ≥ n, and w ∈ B m such that r = I m (w). Then, But, since I n = I m i m−1 i m−2 · · · i n we have Since the monomorphism i m−1 i m−2 · · · i n does not merge conjugacy classes (see [3]) we conclude that x and y are conjugated in B n . We remark the following identity: Lemma 2.8. For η : B ∞ → GL(∞) the Burau representation, the following holds , for all j, k ∈ N.
Proposition 2.9. The sequence r j above is eventually constant and its limit does not depends on the choice of representatives.
Proof. Let N ∈ N be such that N > max{m, k, n} and p and q can be written in the following block configuration: Where d and w are square matrices of dimension N and a and x are square matrices of dimension k. Suppose that for i ≥ N we have r i = r N . We show that r i = r i+1 . As we saw in Proposition 2.2, there are elements u, l ∈ B ∞ such that θ i [k] = uθ i+1 [k]l. Hence, if U = η(u) and L = η(l) we have Furthermore, U and L have the following block configuration Thus, U p = pU and Lq = qL. Consequently, Since Let N > 0 be as before. Consider M > N such that H and J have the block configuration: Where j and h are square matrices of size M . Now, since H preserves the vector v, we have that Therefore we have a well defined product of the double cosets p and q given by Proposition 2.10. The operation defined above is associative. Furthermore, the Burau representation is functor between the categories of double cosets of GL(∞) and of B ∞ .
Proof. The proof of the associative property is analogous to the proof of Theorem 2.6, using Lemma 2.8. The functoriality follows from Lemma 2.8.

Further connexions and generalizations
We can extend the above constructions to the product G [n] = B ∞ × · · · × B ∞ of n copies of the infinite braid group. Let K be the diagonal subgroup of G and let p and q be their respective representatives. Then the operation given by for j sufficiently large, is well defined and associative.
Let ψ : B ∞ → G be a epimorphism and G a group. Let G[α] be the image of B ∞ [α] by ψ, for α ∈ N. Then, the product of double cosets on B ∞ induces a product on the double cosets of G of the form G[α]\G/G[β]. In fact, this fallows from the fact that in the definition of the product of double cosets, the sequence of double cosets defined not only converges, it becomes constant.
For each n ∈ N, consider the symmetric group S n (of the permutations of n elements). If s i is the permutation (i, i + 1) then we have the following presentation Therefore we can regard S n as the quotient group of B n by the relation σ 2 i = 1, 1 ≤ i ≤ n − 1. Let ξ n : B n → S n be the projection map, then this homomorphism gives a correspondence between a braid and the induced permutation of its endpoints. The kernel P n of ξ n is the subgroup of the pure braids in n strands.
As we did for the braid group, consider the direct limit S ∞ of the groups S n with relation to the monomorphisms r n : S n → S n+1 , that take the permutation (k, k + 1) ∈ S n to the permutation (k, k + 1) ∈ S n+1 . Since we have that ξ n r n = i n ξ n+1 , there exists an homomorphism ξ : B ∞ → S ∞ . Using the remarks above, we can define a multiplicative structure on the set of double cosets of S ∞ (in fact, this structure coincide with the one defined by Neretin in [10], [8], [6]). Now, it is easy to see that ξ is an epimorphism.
As a last remark, we point out some similarities between the multiplicative structure defined in B ∞ and that of Aut(F ∞ ). The group Aut(F ∞ ) is defined as follows: Let F n be the free group with n generators x 1 , . . . , x n and denote by Aut(F n ) the group of automorphisms of F n . Then Aut(F ∞ ) = lim Aut(F n ).
The limit is taken with relation to the obvious inclusion Aut(F n ) → Aut(F n+1 ).
For each α ∈ N consider the subgroup H(α) of Aut(F ∞ ) of automorphisms h such that h(x i ) = x i for i ≤ α. In [11], it is defined a product on the double cosets of Aut(F ∞ ) in the following way: Consider the automorphism ϑ j [β] ∈ Aut(F ∞ ) given by Then, for p and q in Aut(F ∞ ), the product of the double cosets H(α)\p/H(β) and H(β)\q/H(γ) is the double coset limit of the sequence pϑ j [m]q in H(α)\Aut(F ∞ )/H(γ).
For each n ∈ N we have a monomorphism i n : B n → Aut(F n ), given by otherwise.
Therefore we can identify B n with the image of i n in Aut(F n ). Consider the limit homomorphism i ∞ : B ∞ → Aut(F ∞ ). The element ϑ j [m] is related to the image of the element θ j [m] as we see in the following proposition Proposition 3.2. Let β be a fixed positive integer. For each k ∈ N, consider y k = x β+k x β+k−1 · · · x β+1 ∈ F ∞ . Then In other words Proof. For k = 1 we have that θ 1 [β] = σ β+1 and therefore We are going to show the truth of the identity by induction on k. Suppose the identity true for k. We can write θ k+1 [β] as θ k+1 [β] = σ k+β+1 · · · σ 2k+β+1 θ k [β]σ 2k+β · · · σ k+β+1 .