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arXiv:math/0609616v2 [math.GT] 22 Feb 2007

Conjugacy in Garside Groups III: Periodic braids

Joan S. Birman∗

Volker Gebhardt Juan Gonz´ alez-Meneses†

February 19, 2007

Abstract

An element in Artin’s braid group Bn is said to be periodic if some power of it lies

in the center of Bn. In this paper we prove that all previously known algorithms for

solving the conjugacy search problem in Bn are exponential in the braid index n for

the special case of periodic braids. We overcome this difficulty by putting to work several

known isomorphisms between Garside structures in the braid group Bnand other Garside

groups. This allows us to obtain a polynomial solution to the original problem in the spirit

of the previously known algorithms.

This paper is the third in a series of papers by the same authors about the conjugacy

problem in Garside groups. They have a unified goal: the development of a polynomial

algorithm for the conjugacy decision and search problems in Bn, which generalizes to other

Garside groups whenever possible. It is our hope that the methods introduced here will

allow the generalization of the results in this paper to all Artin-Tits groups of spherical

type.

1 Introduction

Given a group, a solution to the conjugacy decision problem is an algorithm that determines

whether two given elements are conjugate or not. On the other hand, a solution to the

conjugacy search problem is an algorithm that finds a conjugating element for a given pair

of conjugate elements. In §1.4 of [6] we presented a project to find a polynomial solution to

the conjugacy decision problem and the conjugacy search problem in the particular case of

Artin’s braid group, that is, the Artin-Tits group of type An−1, with its classical or Artin

presentation [1]:

(1)BA

n:

?

σ1,...,σn−1

????

σiσj= σjσi

σiσjσi= σjσiσj

if |i − j| > 1,

if |i − j| = 1.

?

.

One of the steps in the mentioned project asks for a polynomial solution to the above conju-

gacy problems for special type of elements in the braid groups, called periodic braids. This is

achieved in the present paper. More precisely, if we denote by |w| the letter length of a word

w in σ1,...,σn−1and their inverses, we will prove:

∗Partially supported by the U.S.National Science Foundation, under Grant DMS-0405586.

†Partially supported by MTM2004-07203-C02-01 and FEDER.

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Theorem 1. Let wXand wY be two words in the generators σ1,...,σn−1and their inverses,

representing two braids X,Y ∈ BA

n, and let l = max{|wX|,|wY|}. Then there is an algorithm

of complexity O(l3n2logn) which does the following.

(1) It determines whether X and Y are periodic.

(2) If yes, it determines whether they are conjugate.

(3) If yes, it finds a braid C ∈ BA

nsuch that Y = C−1XC.

Here is a guide to this paper. In Section 2, we will review what is known and explain why steps

(1) and (2) of Theorem 1 follow easily from the work in [17, 24, 22]. On the other hand, in

Section 3 we show that the previously known solutions to the conjugacy search problem in the

Artin-Tits group of type An−1present unexpected difficultites, which result in exponential

complexity for periodic braids. Thus they do not meet the requirements of Theorem 1.

A new idea allows us to overcome the difficulty. We have shown that the approach using the

classical Garside structure does not work. The new idea is to put to work the other known

Garside structure on the braid groups and in addition to consider a certain subgroup of the

braid group that arises in the course of our work, and use two known Garside structures on it.

This is accomplished in Section 4, where we give a solution to the conjugacy search problem

for periodic braids which has the stated polynomial complexity. Section 4 divides naturally

into two subsections, according to whether a given periodic braid is conjugate to a power of

δ or ε, two braids that are defined in Section 2 below. The proof in the two cases are treated

in Sections 4.1 and 4.2 respectively. Finally, in Section 5 we compare actual running times of

the algorithms developed in Section 4 to the ones of the best previously known algorithm.

Remark 2. We learned from D. Bessis that he has characterized the conjugacy classes of

periodic elements for all Artin-Tits groups of spherical type. We hope that this characteriza-

tion will allow the generalization of both the techniques and the results of this paper to all

other Artin-Tits groups of spherical type.

Acknowledgements: We are grateful to D. Bessis for useful discussions about his work

in [2] and his forthcoming results, to J. Michel for pointing out that our Corollaries 12 and

15 were known to specialists in Coxeter groups, and also to H. Morton for showing us the

algorithm in [26].

2Known results imply steps (1) and (2) of Theorem 1

Our work begins with a review of known results. Garside groups were introduced by Dehornoy

and Paris in [15]. The main examples of Garside groups are Artin-Tits groups of spherical

type, in particular, Artin braid groups. In this paper we will use two known Garside structures

in the Artin-Tits group of type An−1, and also one Garside structure in the Artin-Tits group

of type Bm.

Although we refer to [6] for a detailed description of Garside structures, we recall here that

such a structure in a group G is given by a lattice order on its elements, together with a

distinguished element of G, called the Garside element, which is usually denoted by ∆. This

partial order and this element ∆ must satisfy several suitable conditions [6].

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The classical Garside structure in the braid groups is related to the presentation (1). The

positive braids are those which can be written as a word in σ1,...,σn−1 (not using their

inverses). The lattice order is defined by saying that X ? Y if X−1Y is a positive braid

(we will say that X is a prefix of Y ). There are special elements called simple braids

which are those positive braids in which any two strands cross at most once. The Gar-

side element ∆ is the positive braid in which any two strands cross exactly once, that is,

∆ = σ1(σ2σ1)(σ3σ2σ1)···(σn−1···σ1). It is also called the half twist, since its geometrical

representation corresponds to a half twist of the n strands. For every braid X ∈ BA

as a word of letter length l, there exists a left normal form, which is a unique way to de-

compose the braid as X = ∆px1···xr, where p is maximal and each xiis a simple braid,

namely the maximal simple prefix of xi···xr. This left normal form can be computed in time

O(l2nlogn) [19].

Artin proved in [1] that the center of BA

nis infinite cyclic and generated by the full twist

∆2= (σ1σ2···σn−1)nof the braid strands. If the braid group is regarded as the mapping

class group of the n-times punctured disc D2

lies in a collar neighborhood of the boundary ∂D2

is said to be periodic if some power of X is a power of ∆2.

n, given

n, then ∆2is a Dehn twist about a curve which

nand is parallel to it. An element X ∈ BA

n

Periodic braids can be thought of as rotations of the disc. Indeed, there is a classical result

by Eilenberg [17] and K´ er´ ekj´ art´ o[24] (see also [12]) showing that an automorphism of the disc

which is a root of the identity (a periodic automorphism) is conjugate to a rotation. Since a

finite order mapping class can always be realized by a finite order homeomorphism [23], this

implies that a periodic braid is conjugate to a rotation. It is not difficult to see that a braid

can be represented by a rotation of D2if and only if it is conjugate to a power of one of the

two braids represented in Figure 1, that is, δ = σn−1σn−2···σ1and ε = σ1(σn−1σn−2···σ1).

(If we need to specify the number of strands, we will write δ = δnand ε = εn.)

Remark 3. The braid ε defined in Figure 1 has a fixed strand, namely strand 2. There are,

to be sure, braids which are conjugate to ε in which the fixed strand is the first one or the last

one, seemingly more natural choices. However, ε is a simple braid, and (as we shall prove in

Proposition 13 below) there is no simple braid which is conjugate to ε and which fixes either

the first or the last strand. This is why we decided to use ε, which fixes the second strand, as

a representative of its conjugacy class. And this is also the reason why, in Section 4.2 below,

we identify the Artin-Tits group of type Bn−1with the subgroup of the n-strand braid group

formed by those braids which fix the second strand, a choice that will surely seem awkward

to specialists.

The theorem of Eilenberg and K´ er´ ekj´ art´ o can then be restated as follows.

Theorem 4. [17, 24] A braid X is periodic if and only if it is conjugate to a power of either

δ or ε.

Notice that δn= εn−1= ∆2. Since ∆2belongs to the center of BA

an efficient algorithm to check whether a braid is periodic.

n, this immediately gives

Corollary 5. A braid X ∈ BA

nis periodic if and only if either Xn−1or Xnis a power of ∆2.

Proof. We only need to prove that the condition is necessary. Suppose that X is periodic.

By Theorem 4, X is conjugate to a power of either δ or ε. In the first case, X = C−1δkC for

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Figure 1: The periodic elements δ and ε.

some C ∈ BA

∆2is central. In the second case, X = C−1εkC, so that Xn−1= C−1εk(n−1)C = C−1∆2kC =

∆2k.

n. Then Xn= C−1δknC = C−1∆2kC = ∆2k, where the last equality holds since

After this result, one can determine whether X is periodic, and also find the power of δ or ε

which is conjugate to X, by the following algorithm.

Algorithm A.

Input: A word w in Artin generators and their inverses representing a braid X ∈ BA

n.

1. Compute the left normal form of Xn−1.

2. If it is equal to ∆2k, return ‘X is periodic and conjugate to εk’.

3. Compute the left normal form of Xn.

4. If it is equal to ∆2k, return ‘X is periodic and conjugate to δk’.

5. Return ‘X is not periodic’.

Proposition 6. The complexity of Algorithm A is O(l2n3logn), where l is the letter length

of w.

Proof. Algorithm A computes two normal forms of words whose lengths are at most nl.

By [19], these computations have complexity O((nl)2nlogn), and the result follows.

We remark that if one knows, a priori, that the braid X is periodic, then one can determine

the power of δ or ε which is conjugate to X by a faster method: Observe that the exponent

sum of a braid X, written as a word in the generators σ1,...,σn−1 and their inverses is

well defined, since the relations in (1) are homogeneous. The exponent sum is furthermore

invariant under conjugacy, hence every conjugate of δkhas exponent sum k(n − 1), whereas

every conjugate of εkhas exponent sum kn. Moreover, the exponent sum determines the

conjugacy class of a periodic braid:

Lemma 7. (Proposition 4.2 of [22]) Let X be a periodic braid. Then X is conjugate to δk

(resp. εk) if and only if X has exponent sum k(n − 1) (resp. kn).

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Computing the exponent sum of a word of length l has complexity O(l). Hence, once it is

known that two given braids are periodic, the conjugacy decision problem takes linear time.

3 Known algorithms are not efficient for periodic braids

We have already determined all conjugacy classes of periodic braids, and we have seen that

the conjugacy decision problem for these braids can be solved very fast. It is then natural to

wonder whether this is also true for the conjugacy search problem. The first natural question

is: Are the existing algorithms for the conjugacy search problem efficient for periodic braids?

The best known algorithm to solve the conjugacy decision problem and also the conjugacy

search problem in braid groups (and in every Garside group) is the one in [21], which consists

of computing the ultra summit set of a braid, defined as follows. Denote by τ the inner

automorphism that is defined by conjugation by ∆. Given Y ∈ BA

is ∆py1···yr, we define its canonical length as ℓ(Y ) = r, and call the conjugates c(Y ) =

∆py2···yrτ−p(y1) and d(Y ) = ∆pτp(yr)y2···yr−1of Y its cycling respectively its decycling.

For every X ∈ BA

n, the ultra summit set USS(X) is the set of conjugates Y of X such that

ℓ(Y ) is minimal and ct(Y ) = Y for some t ≥ 1. It is explained in [21] how the computation

of USS(X) solves the conjugacy decision and search problems in Garside groups.

nwhose left normal form

The complexity of the conjugacy search algorithm given in [21] is proportional to the size

of USS(X), so if one is interested in complexity, it is essential to know how large the ultra

summit sets of periodic braids are. If they turned out to be small, the algorithm in [21] would

be efficient, but we will see in this section that the sizes of ultra summit sets of periodic braids

are in general exponential in n.

More precisely, it was shown by Coxeter in 1934 [13, Theorem 11], that in any finite Coxeter

group, any two elements which are the product of all standard generators, in arbitrary order,

are conjugate. Applied to our case, one sees that the elements of USS(δ) are in bijection

with the elements of the above kind, in the symmetric group Σn. One can count the number

of different elements, and it follows that #(USS(δ)) = 2n−2. The same result is shown in [9,

Chapter V, §6. Proposition 1], in the more general case in which the Coxeter group is defined

by a tree, and also in [29, Lemma 3.2] and in [26, Theorem 2]. Moreover, it can be seen

from the proof in [9] that any two elements in USS(δ) are conjugate by a sequence of special

conjugations, that we denote partial cyclings in [6].

Concerning the elements in USS(ε), in [16, Proposition 9.1] it is shown that any two such

elements are conjugate by a sequence of partial cyclings. It also follows from [16] that every

element in USS(ε) is represented by a word of length n, which is the product of all n − 1

generators, in some order, with one of the generators repeated. One can also count the number

of different elements of this kind, to obtain that #(USS(ε)) = (n − 2)2n−3.

The above arguments show that the sizes of USS(δ) and USS(ε) are exponential with respect

to the number of strands, hence the algorithm in [21] is not polynomial for conjugates of these

braids. In this paper we shall study USS(δ) and USS(ε) in a new way. More precisely, in

Corollaries 12 and 15 we will show that #(USS(δ)) = 2n−2and #(USS(ε)) = (n − 2)2n−3

just by looking at the permutations induced by their elements. This will also provide a fast

solution to the conjugacy search problem in the particular cases of conjugates of δ or ε.

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