Page 1
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.
1
Page 2
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].
2
Page 3
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
3
Page 4
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).
4
Page 5
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.
5
Page 28
Step 3 was performed separately for each sample, first using the algorithm from [21], in the
sequel referred to as Algorithm U, and then again using Algorithm B or Algorithm C. Only
the total time for this step was measured for each case. A memory limit of 512MB and a
time limit of 250s were applied for each test.
All computations were performed on a Linux PC with a 2.4GHz Pentium 4 CPU, 533MHz
system bus and 1.5GB of RAM using the computational algebra system Magma [8]. An
implementation of Algorithm U written in C is part of the Magma kernel; Algorithms B and
C were implemented in the Magma language.
Remark:
mentioned briefly to explain the observed behavior.
One technical aspect of the implementation of Algorithms B and C needs to be
As Algorithms B and C involve mapping a given word, generator by generator, to another
Garside group, a naive implementation of these algorithms will react very sensitively to the
word length of the given element x.
Note, however, that a conjugate y of x having minimal canonical length with respect to the
usual Garside structure, together with a conjugating element, can be computed by iterated
application of cycling and decycling in time O(ℓ3n3logn), where ℓ is the number of simple
factors of x.1Note further that if x is periodic, the canonical length of y as above is at most 1.
Moreover, powers of ∆2can be discarded for the purpose of computing conjugating elements,
as ∆2is central in Bn. The techniques from Algorithms B and C are then applied to the
resulting element whose length in terms of Artin generators is bounded by n2.
While this does not improve the complexity bounds, it significantly reduces computation
times, especially for large values of the parameter c above, and is critical for the cross-over
points between Algorithm U on the one hand and Algorithms B and C on the other hand.
We finally remark that in the special case that the minimal canonical length of conjugates
of x is 0, that is, in the case that x is conjugate to a power of ∆, its ultra summit set has
cardinality 1 and we do not have to use Algorithms B and C, as a conjugating element can
be obtained directly, just by iterated application of cycling and decycling.
The main results can be summarized as follows; see Tables 1 and 2.
1. Time (and memory) requirements of Algorithm U increase rapidly with increasing value
of n. With the exception of elements which are conjugate to a power of ∆, conjugacy
search using Algorithm U fails for n ? 15.
In the light of the exponential growth of USS(δ) and USS(ε) established in Corollar-
ies 12 and 15, this had to be expected.
2. In contrast to this, the computation times for Algorithms B and C grow much more
slowly with increasing value of n. The data is consistent with a polynomial growth;
a regression analysis for fixed values of the parameters k and c suggests that average
times are proportional to nen, where the value en≈ 2.2 is suggested by a regression
analysis.2
1Note that ℓ, unlike the letter length l, is not bounded below by n for periodic braids.
2Note that for fixed values of k and c the word length l is not fixed but grows at least linearly in n;
cf. Lemma 18. Hence this value of en does not contradict the complexity bounds from Propositions 19 and 28.
28
Page 29
Table 1: Total execution times of Algorithms U, B and C for all 100 elements of a sample for
c = 10 and various values of n and k. Where no value is given, either the memory limit of
512MB or the time limit of 250s was exceeded.
k
n
12
57 10
1.56
0.07
4.05
0.23
1520
—
0.34
—
0.97
50
—
3.56
—
6.92
57 10 15
—
0.16
—
0.52
20
—
0.29
—
1.01
6
20
—
0.29
—
1.09
50
—
2.75
—
6.95
U[δ]
B
U[ε]
C
k
n
U[δ]
B
U[ε]
C
0.03
0.02
0.03
0.05
0.12
0.04
0.19
0.12
88.14
0.16
—
0.53
0.02
0.02
0.02
0.01
0.38
0.03
0.16
0.10
22.15
0.06
64.22
0.25
3
15
—
0.12
—
0.60
4
7 1020
—
0.34
—
1.03
50
—
2.79
—
7.02
10
3.86
0.06
0.45
0.22
15
—
0.16
—
0.57
20
—
0.23
—
1.07
50
—
2.37
—
7.02
15
—
0.10
—
0.53
50
—
2.39
—
7.22
0.05
0.04
0.02
0.01
58.81
0.08
9.59
0.22
k
n
789 10 11
50
—
3.18
—
8.23
12
50
—
2.83
—
8.26
15
6.17
0.12
0.09
0.02
20
—
0.33
—
1.06
50
—
3.04
—
7.86
20
—
0.18
—
1.02
50
—
2.60
—
7.68
20
—
0.23
130.34
0.95
50
—
3.03
—
7.84
20
0.16
0.03
67.69
0.73
50
—
1.71
—
7.96
U[δ]
B
U[ε]
C
Table 2: Total execution times of Algorithms U, B and C for all 100 elements of a sample for
c = 250 and various values of n and k. Where no value is given, either the memory limit of
512MB or the time limit of 250s was exceeded.
k
n
12
57 10
2.05
0.67
4.32
0.83
15 20
—
1.83
—
2.37
50
—
8.24
—
10.75
57 10 15
—
1.22
—
1.57
20
—
1.76
—
2.42
50
—
6.79
—
10.69
U[δ]
B
U[ε]
C
k
n
U[δ]
B
U[ε]
C
0.16
0.16
0.16
0.19
0.40
0.32
0.49
0.40
85.20
1.21
—
1.51
0.15
0.16
0.14
0.14
0.65
0.33
0.42
0.38
20.42
0.66
59.76
0.86
346
7 10 15
—
1.14
—
1.59
20
—
1.81
—
2.47
50
—
6.86
—
11.06
10
4.36
0.65
0.99
0.85
15
—
1.22
—
1.60
20
—
1.66
—
2.55
50
—
6.26
—
10.85
15
—
1.11
—
1.57
20
—
1.76
—
2.52
50
—
6.36
—
11.19
0.33
0.31
0.31
0.29
56.14
0.69
9.64
0.83
k
n
789 10 11
50
—
7.00
—
12.51
12
50
—
6.53
—
11.93
15
7.72
1.15
1.04
0.99
20
—
1.89
—
2.50
50
—
6.79
—
11.57
20
—
1.62
—
2.49
50
—
6.37
—
11.47
20
—
1.70
162.83
2.43
50
—
6.80
—
11.55
20
1.44
1.41
90.88
2.21
50
—
5.23
—
11.70
U[δ]
B
U[ε]
C
29
Page 30
In particular, solving the conjugacy search problem for periodic elements using Algo-
rithm D is is not significantly harder than other operations in with braids, that is, it is
feasible whenever the parameter values permit any computations at all.
3. The computation times of Algorithm U depend in a very sensitive way on the value of
k, whereas the running times of Algorithms B and C, with the exception of elements
which are conjugate to a power of ∆ and are treated differently, show relatively little
dependency on k.
4. Average running times for all algorithms appear to be sub-linear in c for fixed values of
the parameters n and k.
For Algorithm U, the effect of c becomes negligible for n ? 10. This is no surprise as
the value of c only affects the initial computation of a conjugate with minimal canonical
length; the time used in this step of the computation is only relevant if the ultra summit
set is small.
5. Using the implementations as explained above, the cross-over point between Algorithm
U and Algorithm B was n ≈ 5, whereas the cross-over point between Algorithm U
and Algorithm C was n ≈ 7; the latter corresponds to the cross-over point between
Algorithm U and Algorithm D for the implementations used in our tests.
We remark that the fact that Algorithms B and C were implemented in the Magma
language (which is partly an interpreter language) incurs some overhead compared to
the C implementation of Algorithm U. This overhead is probably not significant for
Algorithm B, as its implementation is quite simple.3However, for Algorithm C the
overhead can be expected to be significant, as its implementation is rather complex.4
This difference can be assumed to be the main cause for the different cross-over points,
whence a cross-over point of n ≈ 5 for comparable implementations of Algorithms U
and D seems likely.
Remark 30. After this paper was accepted for publication, and as we were preparing this
final copy for the publisher, we learned that E-K Lee and S.J. Lee had posted on the arXiv
their own solution to the same problem [25]. They reference our work and suggest some small
improvements in it.
References
[1] E. Artin, Theorie der Z¨ opfe, Abh. Math. Sem. Hamburg, 4 (1925), 47-72.
[2] D. Bessis, The dual braid monoid, Ann. Sci.´Ecole Norm. Sup. (4) 36 (2003), No. 5,
647-683.
[3] D. Bessis, F. Digne and J. Michel, Springer theory in braid groups and the Birman-Ko-Lee
monoid, Pacific J. Math. 205 (2002) 287310.
3Uses 20 lines of Magma code. As Magma provides a kernel function computing ultra summit sets with
respect to the Birman-Ko-Lee presentation, no low level operations had to be written in the Magma language.
4Uses 200 lines of Magma code. Many low level operations had to be written in the Magma language.
30
Page 31
[4] J. Birman, K. Y. Ko and S. J. Lee, A new approach to the word and conjugacy problems
in the braid groups, Adv. Math. 139, No. 2, (1998), 322-353.
[5] J. Birman, K. Y. Ko and S. J. Lee, The infimum, supremum and geodesic length of a
braid conjugacy class, Adv. Math. (2001), 164, No. 1, (2001), 41-56.
[6] J. Birman, V. Gebhardt and J. Gonz´ alez-Meneses, Conjugacy in Garside groups I: Cy-
cling, Powers and Rigidity, preprint arXiv math.GT/0605230.
[7] J. Birman, V. Gebhardt and J. Gonz´ alez-Meneses, Conjugacy in Garside groups II:
Structure of the Ultra Summit Set, preprint arXiv math.GT/0606652.
[8] W. Bosma, J. Cannon and C. Playoust, The MAGMA algebra system I: The user lan-
guage, J. Symbolic Comput. 24 (1997) 235–265, See also the Magma homepage at
http://magma.maths.usyd.edu.au/magma/.
[9] N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6. Elements of Mathematics
(Berlin). Springer-Verlag, Berlin, 2002.
[10] E. Brieskorn, Die Fundamentalgruppe des Raumes der regul¨ aren Orbits einer endlichen
komplexen Spiegelungsgruppe, Invent. Math. 12 (1971), 57-61.
[11] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17
(1972), 245-272.
[12] A. Constantin and B. Kolev, The theorem of Ker´ ekj´ art´ o on periodic homeomorphisms of
the disc and the sphere, Enseign. Math (2) 40 (1994), 193-204.
[13] H. S. M. Coxeter, Discrete groups generated by reflections. Ann. of Math. (2) 35 (1934),
no. 3, 588–621.
[14] J. Crisp, Injective maps between Artin groups, Geometric group theory down under (Can-
berra, 1996), 119–137, de Gruyter, Berlin, 1999.
[15] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalizations of
Artin groups, Proc. London Math. Soc. 79 (1999), No. 3, 569-604.
[16] F. Digne and J. Michel. Endomorphisms of Deligne-Lusztig varieties. Nagoya Math. J.
183 (2006), 35–103.
[17] S. Eilenberg,
Fund. Math. 22 (1934), 28-41.
Sur les transformations p´ eriodiques de la surface de la sph` ere,
[18] E. ElRifai and H. Morton, Algorithms for positive braids, Quart. J. Math. Oxford Ser
(2), 45 (180) (1994), 479-497.
[19] D. Epstein, J. Cannon, F. Holt, S. Levy, M. Patterson and W. Thurston, Word processing
in groups, Jones and Bartlett, Boston, MA 1992.
[20] F. Garside, The braid group and other groups, Quart. J. Math Oxford 20 (1969), 235-254.
[21] V. Gebhardt, A new approach to the conjugacy problem in Garside groups, Journal of
Algebra 292, No. 1 (2005), 282-302.
31
Page 32
[22] J. Gonz´ alez-Meneses, The nthroot of a braid is unique up to conjugacy, Algebraic and
Geometric Topology 3 (2003), 1103-1118.
[23] S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2,
235–265.
[24] B. de Ker´ ekj´ art´ o,¨Uber die periodischen Transformationen der Kreisscheibe und der
Kugelfl¨ ache, Math. Annalen 80 (1919), 3-7.
[25] E-K LeeandS.J. Lee, Conjugacyclassesof periodicbraids, preprint
arXiv:math.GT/0702349.
[26] H. Morton and R. Hadji, Conjugacy for positive periodic permutation braids, preprint
arXiv math.GT/0312209.
[27] W. Magnus, A. Karass and D. Solitar, Combinatorial Group Theory, 1066, John Wiley
and Sons.
[28] V. Reiner, Non-crossing partitions for classical reflection groups, Discrete Math. 177
(1997) 195-222.
[29] J.-Y. Shi, The enumeration of Coxeter elements. J. Algebraic Combin. 6 (1997), no. 2,
161–171.
Joan S. BirmanVolker Gebhardt Juan Gonz´ alez-Meneses
Departamento de´Algebra,
Universidad de Sevilla,
Apdo. 1160,
41080 Sevilla, Spain.
meneses@us.es
Department of Mathematics,
Barnard College and Columbia University,
2990 Broadway,
New York, New York 10027, USA.
jb@math.columbia.edu
School of Computing and Mathematics,
University of Western Sydney,
Locked Bag 1797,
Penrith South DC NSW 1797, Australia,
v.gebhardt@uws.edu.au
32