Braid Groups

Introduction

The braid group $B_n$ is generated by $n-1$ elements $\sigma_1, \ldots, \sigma_{n-1}$ subject to the braid relations

$$ \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \quad (|i - j| \geq 2), $$

and to no further relations. The presentation differs from that of the symmetric group only in the absence of the relations $\sigma_i^2 = 1$, so $B_n$ is the group obtained from the Coxeter system of type $A_{n-1}$ by deleting the involutions; it maps onto $S_n$ by sending $\sigma_i$ to the adjacent transposition, and the kernel is the pure braid group $P_n$. Despite its elementary presentation, $B_n$ is the meeting point of several theories: it is torsion-free, has a solvable word and conjugacy problem, is left-orderable and linear, and it is the algebraic shadow of the topological theory of braids and of the mapping class group of the disk.

This article is the fifteenth of the corpus and the sixth of the group articles, below the foundational layer and Infinite Abelian Groups, Solvable and Nilpotent Groups, Combinatorial Group Theory, Infinite Groups and Coxeter Groups. It uses the presentation theory and the word problem of Combinatorial Group Theory, the exchange and deletion combinatorics of Coxeter Groups, and the free groups of Generators, Presentations and Free Products. The treatment is algebraic: the group is studied through its presentation, its symmetric-group quotient, its pure subgroup, the Garside normal form, and its order and linearity properties. The geometric realisation of a braid as a system of strands, the mapping class group of a surface, the topological interpretation of the braid group as the fundamental group of a configuration space, and the applications to knot theory all belong to Part II, where a space and a distance are available, and they are deferred.

The Braid Group and its Presentation

Artin's Presentation

Definition. For $n \geq 2$ the braid group on $n$ strands is

$$ B_n = \left\langle \sigma_1, \ldots, \sigma_{n-1} \mid \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}\ (1 \leq i \leq n-2),\ \ \sigma_i \sigma_j = \sigma_j \sigma_i\ (|i - j| \geq 2)\right\rangle, $$

with $B_1$ the trivial group. The generators $\sigma_i$ are the Artin generators, and the two families of relations are the braid relations.

Proposition. $B_n$ is finitely presented, and the presentation above is the Coxeter presentation of the symmetric group with the relations $\sigma_i^2 = 1$ removed. The braid relations are exactly the Coxeter relations $(s_is_{i+1})^3 = 1$ and $(s_is_j)^2 = 1$ written without the extraneous letters: in a group in which $\sigma_i^2 = 1$, the relation $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ is equivalent to $(\sigma_i\sigma_{i+1})^3 = 1$.

Proof. The first statement is clear from the presentation. For the second, multiply the braid relation on the left by $\sigma_i$ and on the right by $\sigma_{i+1}$ and use the involutions: $\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}$ becomes, after left multiplication by $\sigma_i$ and right by $\sigma_{i+1}$, the identity $\sigma_{i+1}\sigma_i = \sigma_i\sigma_{i+1}\sigma_i\sigma_{i+1}$, and the reverse computation gives $(\sigma_i\sigma_{i+1})^3 = 1$ when $\sigma_i^2=\sigma_{i+1}^2=1$.

Definition. The Artin–Tits group (or Artin group) of a Coxeter system $(W,S)$ is the group with the same presentation as $W$ but omitting the relations $s^2 = 1$; $B_n$ is the Artin group of type $A_{n-1}$. The class of Artin groups is the natural home of the braid group; the finite-type ones (those whose Coxeter diagram is finite) are called spherical, and the braid groups are the spherical Artin groups of type $A$.

The Symmetric-Group Quotient

Theorem. Let $s_i$ be the adjacent transposition $(i\ i+1)$ in $S_n$. The assignment $\sigma_i \mapsto s_i$ extends to a surjective homomorphism

$$ \pi : B_n \longrightarrow S_n, $$

whose kernel is the pure braid group $P_n$. Hence $P_n \trianglelefteq B_n$ and $B_n/P_n \cong S_n$, with $[B_n : P_n] = n!$.

Proof. The transpositions $s_i$ satisfy the braid relations — this was verified in the discussion of type $A$ in Coxeter Groups, where all the defining relations of the symmetric group as a Coxeter group were checked — so the assignment extends to a homomorphism by the universal property of the presentation. It is surjective because the $s_i$ generate $S_n$. The kernel is $P_n$ by definition, and the index is the order of the image.

Proposition. The extension $1 \to P_n \to B_n \to S_n \to 1$ does not split for $n \geq 2$: there is no homomorphism $S_n \to B_n$ whose composite with $\pi$ is the identity.

Proof. Such a section would send a transposition, an element of order $2$, to an element of order $2$ in $B_n$. But $B_n$ is torsion-free, as proved below, so it has no element of order $2$.

The quotient $B_n \to S_n$ is the algebraic content of the statement that a braid determines a permutation of its strands, the permutation being the one induced on the endpoints; the kernel $P_n$ consists of the braids whose strands return to their starting points.

The Pure Braid Group

Artin's Generators

Definition. For $1 \leq i < j \leq n$ let

$$ A_{ij} = \sigma_{j-1} \sigma_{j-2} \cdots \sigma_{i+1}\, \sigma_i^2\, \sigma_{i+1}^{-1} \cdots \sigma_{j-2}^{-1} \sigma_{j-1}^{-1} \in P_n. $$

The elements $A_{ij}$ are the Artin generators of the pure braid group.

Theorem (Artin). $P_n$ is generated by the $A_{ij}$ for $1 \leq i < j \leq n$, and it has a presentation on these generators whose relations are of two kinds: the commutation relations

$$ A_{ij} A_{kl} = A_{kl} A_{ij} \qquad (i < j < k < l \text{ or } i < k < l < j), $$

for pairs of indices that are disjoint or nested, together with, for each interleaved pair $i < k < j < l$, a relation expressing $A_{ij}A_{kl}A_{ij}^{-1}$ as an explicit product of generators with indices among $i, j, k, l$. The full list is Artin's pure braid presentation, and the subgroup presented is the kernel of $B_n \to S_n$.

The Recursive Structure

Theorem. For $n \geq 2$ there is a split short exact sequence

$$ 1 \longrightarrow F_{n-1} \longrightarrow P_n \longrightarrow P_{n-1} \longrightarrow 1, $$

where $F_{n-1}$ is the free group of rank $n-1$; the splitting exhibits $P_n$ as a semidirect product $P_n \cong F_{n-1} \rtimes P_{n-1}$. Consequently $P_n$ has a normal series with free factors, is torsion-free, and is residually a torsion-free nilpotent group.

Proof sketch. The map $P_n \to P_{n-1}$ deletes the last strand, and its kernel is the subgroup generated by the $A_{in}$ for $i = 1, \ldots, n-1$, which is free of rank $n-1$ because the $A_{in}$ satisfy no relation among themselves; the image is $P_{n-1}$ and the extension splits because the braids acting only on the first $n-1$ strands furnish a complement. The residual nilpotence follows by iterating the semidirect decomposition, since an extension of a free group by a residually torsion-free nilpotent group, with the action preserving the filtration, is again residually torsion-free nilpotent.

Corollary. $P_n$ is torsion-free, its abelianisation is free abelian of rank $\binom{n}{2}$, and it is residually torsion-free nilpotent and hence residually finite. In particular $P_2 \cong \mathbb{Z}$ and $P_3 \cong F_2 \times \mathbb{Z}$: $P_2$ is generated by $A_{12}$, and $P_3$ is the direct product of the free group on $A_{13}, A_{23}$ with the central subgroup generated by the full twist.

Torsion and Central Elements

Theorem. $B_n$ is torsion-free for every $n$.

Proof sketch. Let $w \neq 1$ have Garside normal form $\Delta^m a_1\cdots a_k$ with $k \geq 1$ and $a_k \neq 1$. Computing the normal form of $w^N$ by iterating the greedy algorithm shows that the number of simple factors grows with $N$, so the normal form of $w^N$ never becomes a power of $\Delta$; hence $w$ has infinite order. The only remaining case $w = \Delta^m$ is also of infinite order unless $m = 0$.

Theorem (centre). For $n \geq 3$ the centre of $B_n$ is infinite cyclic, generated by the square of the Garside element

$$ \Delta = (\sigma_1\sigma_2\cdots\sigma_{n-1})(\sigma_1\sigma_2\cdots\sigma_{n-2})\cdots(\sigma_1\sigma_2)(\sigma_1) \in B_n, $$

so $Z(B_n) = \langle \Delta^2\rangle \cong \mathbb{Z}$; $B_2 \cong \mathbb{Z}$ with centre $B_2$; and for $n \geq 3$ the centre of $P_n$ is also generated by $\Delta^2$.

Proof sketch. The element $\Delta$ is the product of the prefix braids $\sigma_1\sigma_2\cdots\sigma_k$ for $k = n-1, n-2, \ldots, 1$ and is the Garside element; its square is central: $\Delta^2$ commutes with every $\sigma_i$, which is checked from the braid relations, and it is the least positive central element. Conversely a central element of $B_n$ has a Garside normal form whose simple factors are permuted cyclically by conjugation by $\Delta$, so centrality forces all but a power of $\Delta^2$ to vanish. The detailed argument is Garside's. The verification accompanying this article confirms the centrality of $\Delta^2$ in $B_3$ using a faithful matrix representation, and the general statement is standard.

Corollary. $B_3 / Z(B_3) \cong \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$ and $P_3/Z(P_3) \cong F_2$; the quotient of $B_3$ by its centre is the free product of cyclic groups of orders $2$ and $3$, the modular group presented as a free product.

Algebraic Properties and the Word Problem

The Garside Normal Form

Definition. A braid is positive (or a positive word) if it is a product of Artin generators with positive exponents. The simple elements of $B_n$ are the positive elements that divide $\Delta$, that is, the elements $a$ for which there is a positive $b$ with $ab = \Delta$; they correspond to the elements of $S_n$ under the quotient map $B_n \to S_n$, so there are $n!$ of them, and $\Delta$ itself is the largest simple element, being the positive braid that maps to the longest element of $S_n$.

Theorem (Garside). Every element $w \in B_n$ has a unique normal form

$$ w = \Delta^{m} \, a_1 a_2 \cdots a_k, $$

where $m \in \mathbb{Z}$, each $a_i$ is a simple element, $a_k \neq 1$, and the decomposition is the greedy one: the factors are chosen successively as the largest possible simple elements, in the sense of the left-divisor conditions for the divisibility order on positive braids. The normal form is computed effectively, so the word problem of $B_n$ is solvable; the conjugacy problem is also solvable, and $B_n$ is biautomatic.

Proof sketch. The existence of the greedy decomposition rests on the fact that every positive element $w$ has a unique maximal simple divisor, and the algorithm peels off the maximal simple elements successively. Termination follows from the decrease of the infimum of the prefixes, and the normal form is unique because the greedy choice is forced. Garside's algorithm is the first solution of the word problem for the braid groups; the conjugacy problem was solved by Garside and by ElRifai–Morton with the same combinatorial machinery, and the biautomatic structure gives another proof of both.

Definition. The Garside element $\Delta$ satisfies $\Delta \sigma_i \Delta^{-1} = \sigma_{n-i}$ for every $i$; it induces the reversal permutation on the strands, and $\Delta^2$ generates the centre.

Linearity and Orderability

Theorem. $B_n$ is linear: for every $n$ there is a faithful representation of $B_n$ into a general linear group over a ring of Laurent polynomials, and hence $B_n$ is residually finite; consequently $B_n$ has a solvable word problem for a second reason and is hopfian.

Proof sketch. The representation is the Lawrence–Krammer–Bigelow representation, which sends each Artin generator to an explicit matrix over a ring of Laurent polynomials in one variable. Faithfulness was proved by Bigelow and by Krammer independently; the representation extends the classical Burau representation, which is faithful for $n \leq 3$, is not faithful for $n \geq 5$, and is not settled for $n = 4$.

Theorem (Dehornoy). $B_n$ is left-orderable: there is a total order on $B_n$ invariant under left multiplication. Consequently $B_n$ contains no element of finite order other than the identity, has no nontrivial finite normal subgroup, and the ordering provides a normal form and a comparison algorithm for the word problem.

Proof sketch. Dehornoy's ordering is defined by a $\sigma$-positive form: a braid is $\sigma$-positive if there is an index $i$ such that it admits an expression in the generators $\sigma_j^{\pm 1}$ with $j \neq i$ together with $\sigma_i$ alone, the inverse $\sigma_i^{-1}$ not occurring; every nonidentity braid is $\sigma$-positive or has a $\sigma$-positive inverse, and one declares $A < B$ when $A^{-1}B$ is $\sigma$-positive. The transitivity and left-invariance are verified by a case analysis on such forms, and the order is known as the sigma-ordering.

Remark. Being left-orderable, $B_n$ cannot have a nontrivial finite normal subgroup, and the quotient maps to finite groups are as large as they can be; the combination of left-orderability, bi-automaticity, residual finiteness and linearity makes $B_n$ one of the most rigid infinite groups known, in marked contrast with the pathological torsion groups of Infinite Groups.

The Positive Braid Monoid

Fractions and Generation

Definition. The positive braid monoid $B_n^+$ is the monoid with generators $\sigma_1,\ldots,\sigma_{n-1}$ and the braid relations; its elements are the positive braids. The exponent sum $\sigma_i \mapsto 1$ defines a homomorphism $B_n \to \mathbb{Z}$, the exponent sum, which is surjective.

Proposition. $B_n^+$ embeds in $B_n$ and is cancellative on both sides; and $B_n$ is its group of fractions: every element of $B_n$ is of the form $ab^{-1}$ with $a, b \in B_n^+$.

Proof sketch. The defining relations of the group are length-preserving in the positive generators and never introduce an inverse generator, so a positive word that represents the identity in $B_n$ is trivial in the monoid; this gives the embedding. Cancellativity follows because $B_n$ is a group and $B_n^+$ embeds in it, so a cancellation $ac = bc$ in $B_n^+$ holds in $B_n$ and gives $a = b$. For the fractions, every positive element divides a power of the Garside element $\Delta$ — the greedy algorithm of the normal form exhibits the quotient as a positive braid — so given $a, b$ one chooses powers of $\Delta$ divisible by $a$ and by $b$ and rebalances to produce a common right multiple; this is the Ore condition.

Proposition. $B_n$ is generated by two elements: the Artin generator $\sigma_1$ and the prefix braid $\delta = \sigma_1\sigma_2\cdots\sigma_{n-1}$. Its abelianisation is $\mathbb{Z}$, generated by the image of any one Artin generator, and the exponent sum is the abelianisation map.

Proof. The braid relations give $\delta \sigma_1 \delta^{-1} = \sigma_2$, and more generally $\delta \sigma_i \delta^{-1} = \sigma_{i+1}$ for $i \leq n-2$; conjugating $\sigma_1$ by powers of $\delta$ therefore produces every Artin generator, so $\sigma_1$ and $\delta$ generate. For the abelianisation, the relations force all the $\sigma_i$ to be equal in any abelian quotient, and the exponent sum realises this quotient.

Garside Monoids

Definition. A Garside monoid is a cancellative monoid $M$ with a Garside element $\Delta$ such that the left and right divisors of $\Delta$ coincide, are finite in number, generate $M$, and the conjugation action of $\Delta$ on the divisors is a bijection; the divisors are the simple elements. The braid monoid $B_n^+$ is a Garside monoid with the simple elements of the previous sections, and its group of fractions is $B_n$.

Theorem. The class of Garside monoids is closed under the constructions that produce the spherical Artin–Tits monoids: the positive monoids of spherical type are Garside monoids, and their groups of fractions are the corresponding Artin groups. In each case the simple elements form a finite lattice, the word problem is solved by the greedy normal form, and the group is torsion-free.

The Garside framework is the abstract statement of the combinatorics used for $B_n$: it isolates the properties of $\Delta$ and of the simple elements that make the normal form work, and it applies to the spherical Artin groups of types $B$, $D$ and the exceptional types as well as to type $A$.

Small Cases and Examples

The Cases $n = 2$ and $n = 3$

Proposition. $B_1 = 1$ and $B_2 = \langle \sigma_1\rangle \cong \mathbb{Z}$. Also $P_1 = 1$, $P_2 = B_2 \cong \mathbb{Z}$.

Proposition. $B_3 = \langle \sigma_1, \sigma_2 \mid \sigma_1\sigma_2\sigma_1 = \sigma_2\sigma_1\sigma_2\rangle$ is the amalgamated free product of two infinite cyclic groups over a subgroup of index $2$ in one factor and index $3$ in the other; equivalently $B_3 \cong \langle a, b \mid a^2 = b^3\rangle$, the trefoil group. Its centre is generated by $\Delta^2 = (\sigma_1\sigma_2)^3$, and $B_3/Z(B_3) \cong \mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$.

Proof sketch. Set $a = \sigma_1\sigma_2\sigma_1$ and $b = \sigma_1\sigma_2$; then $a^2 = (\sigma_1\sigma_2)^3 = b^3$, giving the presentation $\langle a,b\mid a^2=b^3\rangle$, and $\langle a\rangle \cap \langle b\rangle = \langle a^2\rangle$ is infinite cyclic of index $2$ in $\langle a\rangle$ and index $3$ in $\langle b\rangle$, so the group is the amalgam of the two infinite cyclic groups over that subgroup. The centre is the cyclic subgroup generated by the common value $a^2 = b^3$. The quotient by the centre is generated by the images of $a$ and $b$, which have orders $2$ and $3$ and generate freely as a free product, using that a relation between them would lift to a relation modulo the centre and hence to a power of the central element.

Example. The braid group $B_3$ is the simplest braid group that is not abelian and not free; it is an extension $1 \to P_3 \to B_3 \to S_3 \to 1$ with $P_3 \cong F_2 \times \mathbb{Z}$, and its centre is infinite cyclic. The verification accompanying this article computes the reduced Burau representation of $B_3$, checks the braid relation in it, and confirms that $\Delta^2$ is central.

Automorphisms

Theorem (Dyer–Grossman). For $n \geq 3$ the automorphism group of $B_n$ is generated by the inner automorphisms together with the reversal automorphism $\sigma_i \mapsto \sigma_{n-i}$ and the inversion automorphism $\sigma_i \mapsto \sigma_i^{-1}$.

Proof sketch. The action of an automorphism on the abelianisation $\mathbb{Z}$ and on the symmetric-group quotient $S_n$ is computed first; the reversal and inversion realise the two possible nontrivial actions, and the kernel of the map to these data is the group of automorphisms acting trivially, which is shown to consist of inner automorphisms by an analysis of the images of the generating set.

Corollary. For $n \geq 3$ the centre of $B_n$ is characteristic and the outer automorphism group $\mathrm{Out}(B_n)$ is finite; the rigidity of the automorphism group, together with the order structure, is the reason the braid groups are the model spherical Artin groups for the theory of automorphisms.

Remark. The braid group is the algebraic form of the mapping class group of a disk with marked points, and the presentation above is the algebraic skeleton of the topological theory; the identification of $B_n$ with that mapping class group, the classification of braids up to isotopy, and the application to knots and links are developed in Part II, where the configuration space and its fundamental group are available. The present article treats $B_n$ as the finitely presented group of the braid relations.

Summary

The braid group $B_n$ has Artin's presentation with generators $\sigma_1,\ldots,\sigma_{n-1}$ and the braid relations, the symmetric-group relations with the involutions removed; it is the Artin group of type $A_{n-1}$. The assignment $\sigma_i \mapsto (i\ i+1)$ gives a surjection $B_n \to S_n$ with kernel the pure braid group $P_n$ of index $n!$, and the extension does not split because $B_n$ is torsion-free. The pure braid group is generated by the elements $A_{ij}$ for $i < j$, with Artin's presentation, and the recursive decomposition $P_n \cong F_{n-1} \rtimes P_{n-1}$ makes $P_n$ residually torsion-free nilpotent and torsion-free; $P_2 \cong \mathbb{Z}$ and $P_3 \cong F_2 \times \mathbb{Z}$.

For $n \geq 3$ the centre of $B_n$ is the infinite cyclic group generated by $\Delta^2$, and the centre of $P_n$ is also generated by $\Delta^2$; $B_2 \cong \mathbb{Z}$ and $B_1$ is trivial. Garside's normal form $w = \Delta^m a_1\cdots a_k$ over the simple elements gives the word problem, and the conjugacy problem and biautomaticity follow from the same combinatorics. The braid groups are left-orderable by Dehornoy's ordering, linear by the Lawrence–Krammer–Bigelow representation, residually finite and hopfian. $B_3$ is the trefoil group $\langle a,b \mid a^2 = b^3\rangle$ with $B_3/Z(B_3) \cong \mathbb{Z}/2 * \mathbb{Z}/3$, and the automorphisms of $B_n$ for $n \geq 3$ are generated by the inner automorphisms, the reversal and the inversion. The topological theory of braids, mapping class groups and knots belongs to Part II.

Summary of Notation

Symbol	Meaning
$B_n$	Braid group on $n$ strands
$\sigma_1,\ldots,\sigma_{n-1}$	Artin generators
$P_n$	Pure braid group, the kernel of $B_n \to S_n$
$A_{ij}$	Artin generator of $P_n$, $1 \leq i < j \leq n$
$\Delta$	Garside element (half twist)
$\Delta^2$	Full twist, generator of the centre for $n \geq 3$
$F_{n-1}$	Free group of rank $n-1$
$S_n$	Symmetric group, the quotient $B_n/P_n$
$a_1\cdots a_k$	Simple factors of the Garside normal form
$\mathbb{Z}/2 * \mathbb{Z}/3$	Free product, the quotient $B_3/Z(B_3)$