Regular homotopy of Hurwitz curves

0.33 MB·

by  Denis Auroux

REGULAR HOMOTOPY OF HURWITZ CURVES
4
0 DENIS AUROUX, VIKTOR S. KULIKOV,
0 AND VSEVOLOD V. SHEVCHISHIN
2

n
Abstract. We prove that any two irreducible cuspidal Hurwitz
a
J curves C0 and C1 (or more generally, curves with A-type singu-
4 larities) in the Hirzebruch surface FN with coinciding homology
1 classesandsetsofsingularitiesareregularhomotopic;andsymplec-
tically regularhomotopicif C0 andC1 aresymplecticwith respect
]
to a compatible symplectic form.
G
S
.
h
t
a
0. Introduction
m
[ In this paper, we deal with J-holomorphic curves in the projective

1 plane and Hurwitz curves (in particular, algebraic curves) in the Hirze-
v bruch surfaces F which imitate the behavior of plane algebraic curves
2 N
7 withrespect to pencils of lines (thedefinition ofHurwitz curves isgiven
1
in Section 2). We restrict ourselves to the case when Hurwitz curves
1
0 can have only singularities of the types An with n > 0 (i.e., which
4 are locally given by y2 = xn+1) and also so-called negative nodes (see
0
/ Section 2).
h
t In [Moi] Moishezon proved the existence of an infinite sequence H¯i
a ⊂
m F1 of generic irreducible cuspidal Hurwitz curves of degree 54 with
exactly 378 cusps and 756 nodes which have pairwise distinct braid
:
v
monodromy type. In particular, they are pairwise non-isotopic, and
i
X
almost all of them are not isotopic to an algebraic cuspidal curve.
r
a The aim of this article is to prove the following statement.
Theorem 0.1. Any two irreducible cuspidal Hurwitz curves H¯ and H¯
0 1
in the Hirzebruch surface F having the same homology class and the
N
same numbers of cusps and nodes (or, in presence of negative nodes,
differences between numbers of positive and negative nodes) can be con-
nected by a regular homotopy.
Date: This version: November 2003.
This work was initiated during a stay of the authors at IPAM, UCLA. The first
authorwassupportedinpartbyNSF grantDMS-0244844. The secondauthorwas
supported in part by RFBR (No. 02-01-00786)and INTAS (No. 00-0269).
1
2 D. AUROUX,VIK.S. KULIKOV,AND V.SHEVCHISHIN
Moreover, if H¯ and H¯ are symplectic with respect to some form ω
0 1
compatible with the canonical ruling of F , then the regular homotopy
N
between them can be made ω-symplectic.
¯
A regular homotopy is a deformation family H which is an
t t∈[0,1]
{ }
isotopy except for a finite number of values of t at which the curve
undergoes the “standard” transformation of creation or cancellation of
a pair of nodes of opposite signs, see Section 2 for the precise definition.
The result remains true if the irreducible Hurwitz curves H¯ and H¯
0 1
are allowed to present arbitrary singularities of type A . The necessary
n
andsufficient conditionfortheexistenceofaregularhomotopybetween
H¯ and H¯ then becomes that the numbers of singularities of each type
0 1
arethesame(except inthecase ofnodes, forwhich oneshould compare
differences between numbers of positive and negative nodes).
As a corollary of our main theorem, we obtain:
Corollary 0.2. Let C and C be two ordinary cuspidal irreducible
0 1
symplectic surfaces in (CP2,ω), degC = degC , pseudoholomorphic
0 1
with respect to ω-tamed almost-complex structures J and J respec-
0 1
tively. If C and C have the same numbers of cusps and nodes, then
0 1
they can be deformed into each other by a C1-smooth symplectic regular
homotopy in CP2.
The structure of the rest of this paper is as follows: in Section 1 we
give a symplectic isotopy result for curves in Hirzebruch surfaces whose
irreducible components are sections. In Section 2 we define Hurwitz
curves and regular homotopy, and mention some of their elementary
properties. In Section 3 we introduce the main ingredient in the proof
of Theorem 0.1, namely braid monodromy factorizations of Hurwitz
curves with A-type singularities. The main results are then proved in
Section 4, where the outline of a more geometric alternative proof is
also given.
Finally, in Section 5, we construct two so-called quasi-positive fac-
torizations of an element in the braid group Br which give a negative
4
answer to the Generalized Garside Problem asking whether the natu-
ral homomorphism from the semigroup of quasi-positive braids to the
braid group is an embedding.
Acknowledgement. The authors would like to express their gratitude
to the Institute for Pure and Applied Mathematics at UCLA for its
hospitality during the early stages of the preparation of this paper.
REGULAR HOMOTOPY OF HURWITZ CURVES 3
1. Symplectic isotopy of sections
Let X be a Hirzebruch surface F , k > 0, and E a rational holomor-
k
phic curve with self-intersection E2 = k, which is unique if k > 1.
−
Furthermore, let ω be a K¨ahler form compatible with the complex
structure JF . It is known (see [Li-Liu] or [La-McD]) that (X,ω) is
k
symplectomorphic to F if k is odd or to F otherwise, equipped with
1 0
an appropriate K¨ahler structure.
Let be the set of all Cl-smooth ω-tame almost complex structures
J
on X with a fixed sufficiently large non-integer l. This is a Banach
manifold. Let be the total moduli space of pseudoholomorphic
k
M
curves C on X in the homology class [E] H2(X,Z). Recall that
k
∈ M
is defined as the quotient
Mk := (cid:26)(u,J) : Jmo∈rpJhi,c aun:dPr1ep→resXentiss [JE-]holo-(cid:27) PGL(2,C),
.
where the group PGL(2,C) acts by holomorphic automorphisms on
P1. By abuse of notation, we denote the points of by (u,J) or by
k
M
(C,J),emphasizing eithertheparameterizingmapuortheimagecurve
C. The genus formula for pseudoholomorphic curves (see [Mi-Wh])
ensures that all curves in are embedded. Let pr : be
Mk k Mk → J
the natural projection given by pr : (C,J) J. This is a Fredholm
k
7→
operator of R-index 2(1 k).
−
Lemma 1.1. The projection pr : is
k Mk → J
(1) an embeddingof real codimension2(k 1)if k > 1; in particular,
−
it is an open embedding for k = 1;
(2) a bundle with fiber S2 over an open subset if k = 0.
reg
J ⊂ J
Proof. The properties of the projection pr at (C,J) can be described
k
in terms of the normal sheaf of C (see [Iv-Sh-1] or [Iv-Sh-2]). Since
C
N
C is embedded, reduces to a line bundle, denoted by N , equipped
C C
N
with the Gromov operator D : L1,p(C,N ) Lp(C,N Λ(0,1)).
C,J C C
→ ⊗
Since in our case C is rational, the operator D cannot have both
C,J
non-trivialkernel andcokernel, see[H-L-S], [Iv-Sh-1], or[Iv-Sh-2]. This
means that pr is of “maximal rank” everywhere on , i.e., either an
k Mk
immersion, or a submersion, or a local diffeomorphism.
The global injectivity of pr in the case k > 1 follows easily from
k
the fact that two distinct J-holomorphic curves C and C′ must have
positive intersection number, which wouldcontradict thecondition[C]
·
[C′] = k < 0.
−
4 D. AUROUX,VIK.S. KULIKOV,AND V.SHEVCHISHIN
In the case k = 0 we first show that every non-empty fiber of
pr is compact. Assuming the contrary, we obtain a sequence of J-
0
holomorphiccurvesC withnolimitsetinpr−1(J). Applying Gromov’s
ν 0
compactness theorem we may assume that the sequence converges to
some reducible curve C∗ = m C∗. Here the image is understood in
i i
the sense of cycles. Since tPhe intersection number of every C∗ with
i
every curve in pr−1(J), say C , is non-negative, and the sum of these
0 1
indices is zero, we conclude that [C∗] [C ] = 0 for every i.
i · 1
Therefore each C∗ represents a homology class which is a positive
i
integer multiple of the primitive class [C ]. Counting the ω-area of
1
C∗’s we conclude that C∗ consists of a unique non-multiple component
i
homologous to C . Thus (C∗,J) lies in .
1 0
M
To show that all non-empty fibers of pr are diffeomorphic, it is
0
sufficient to prove that the complement to the image of pr has real
0
codimension 2. Using Gromov’s compactness once more, we can find a
reducible J-holomorphic curve C∗ = m C∗ for every J lying on the
i i
(topological) boundary of the image Pof pr . By the genus formula, all
0
C∗ must be embedded rational curves. Moreover, since c (X) [C∗] = 2
i 1 ·
and the intersection form of F = S2 S2 is even, for one of them,
0
×
say C∗, we must have c (X) [C∗] 6 0. Applying the genus formula
1 1 · 1
we obtain that [C∗]2 =: k 6 2. This implies that J lies in the image
1 −
of the projection pr : , which is locally a submanifold of
k Mk → J
codimension > 2. (cid:3)
We shall denote the image pr ( ) by . For k > 1 this is a
k k k
M J
submanifold of of real codimension 2(k 1), in general not closed.
J −
Lemma 1.2. Let X be a Hirzebruch surface F , ω a K¨ahler form
k
on X, J an almost-complex structure, E the corresponding J-
k
∈ J
holomorphic curve with self-intersection k, and C an irreducible J-
−
holomorphic curve different from E. Let also F be a fiber of the ruling
on X. Then
(i) C is homologous to d[E]+f [F] with d > 0 and f > kd;
(ii) c (X) C = d(2 k)+2f > 0.
1
· −
Proof. Fix a point x on X outside E. Observe that there exists a path
h : [0,1] , t [0,1], which connects J =: J with the “standard”
0
→ J ∈
structure J = J through almost-complex structures J = h(t), such
1 st t
that E is J -holomorphic for every t [0,1]. This means that h takes
t
∈
values in . Consider the moduli space :=
k h
J M
t [0,1], C isarationalJ -holomorphiccurve
Mh(X,F,x) := (C,t) : ho∈mologous to F which ptasses through x.
n o
REGULAR HOMOTOPY OF HURWITZ CURVES 5
together with the natural projection pr : [0,1] given by pr :
h Mh → h
(C,t) t. The techniques of [Bar-1] and [Bar-2] ensure that pr :
h
7→
[0,1] is a diffeomorphism. This implies that there exists a
h
M →
J-holomorphic curve C isotopic to F.
0
ThehomologygroupH (X,Z)isafreeabeliangroupwithgenerators
2
E and F, so [C] = d[E]+f[F] with some integers d and f. Since C,
E, and C are J-holomorphic, the intersection indices [C] [C ] and
0 0
·
[C] [E] are non-negative (see [Mi-Wh]). This gives the inequalities in
·
(i). The equality in part (ii) follows from formulas c (X) E = 2 k
1
· −
and c (X) F = 2. Finally, d(2 k) + 2f = 2d + f + (f kd) is
1
· − −
non-negative and vanishes only if d = f = 0. (cid:3)
Theorem 1.3. Let X be a Hirzebruch surface F , ω , t [0,1] a
k t
∈
smooth family of symplectic forms on X taming the complex structure
JF , E a holomorphic curve with self-intersection k, F a fiber of
k −
the ruling of X, and C an immersed ω -symplectic surface in X such
0 0
that every component C of C is homologous to either to [F] or to
0,i 0
[E]+f [F] with f > 1. Assume also that the surface C E has only
i i 0
∪
positive transversal double points as singularities. Then there exists an
ω -symplectic isotopy C between C and a holomorphic curve C , such
t t 0 1
that each C meets E transversally with positive intersection index.
t
The case of main interest is when all f = k. Then C consists of sec-
i 0
tions of the line bundle X E ∼= P1(k). The general case corresponds
\ O
to a collection of meromorphic sections with various numbers of poles.
Proof. It follows from the assumption of the theorem that both C and
0
E are J -holomorphic curves with respect to a same ω -tame almost
0 0
complex structure J on X (see e.g. [Iv-Sh-1]). Furthermore, there
0
exists a path h : [0,1] , t [0,1], connecting J with the “stan-
0
→ J ∈
dard” structure J = J , such that each J := h(t) is ω -tame and E is
1 st t t
J -holomorphic for every t [0,1]. This means that h takes values in
t
∈
. We assume that h is chosen generic enough.
k
J
By Lemma 1.2, f > k and c (X) [C ] = 2 + 2f k is strictly
i 1 0,i i
· −
positive. On each component C we fix p := c (X) [C ] 1 =
0,i i 1 0,i
· −
1 + 2f k points x ,...x in generic position. Let x = x
i − i,1 i,pi { i,j}
be the whole collection of these points. Consider the moduli space
= (X,C ,x) of deformations of C as J -holomorphic curves
h h 0 0 t
M M
which have the same constellation as C , i.e. such that
0
C has the same number of irreducible components as C ;
0
•
each component C of C is rational and homologous to the cor-
i
•
responding component C of C ;
0,i 0
6 D. AUROUX,VIK.S. KULIKOV,AND V.SHEVCHISHIN
thecomponentC ofC passesthroughthesamepointsx ,...x
• i i,1 i,pi
as C does.
0,i
Now, the moduli space is defined as
Mh(X,C0,x) := (C,t) : ctu∈rve[0w,1it],hCtheiscoansJtte-lhlaotloiomnoorpfhCic .
n 0o
We denote by pr : [0,1] the natural projection given by pr :
h h h
M →
(C,t) t. By abuse of notation we write C meaning that (C,t)
h
7→ ∈ M
lies in for some t.
h
M
The expected real dimension of is 1. The possibility to de-
h
M
form the structures h(t) arbitrarily near the fixed points x ensures the
transversality property of the deformation problem. So is a mani-
h
M
fold of the expected dimension. An important observation of [H-L-S],
see also [Bar-1] and [Bar-2], is that in this situation, because the curves
C are rational, the projection pr has no critical points.
h
So the statement of the theorem would follow from the properness
of pr : [0,1]. Assuming the contrary, we would find a sequence
h h
M →
t converging to some t∗ [0,1] and a sequence of J -holomorphic
n ∈ tn
curves C with no accumulation points in . By Gromov’s
n h h
∈ M M
compactness theorem, going to a subsequence we may assume that C
n
weakly converges to some J∗-holomorphic curve C∗ with J∗ = h(t∗).
Since it is possible to consider the behavior of the components of C
n
separately, we may assume that C and every C are irreducible. Let
0 n
C∗ = l m C∗ be the decomposition of C∗ into irreducible com-
j=1 j j
ponentPs, mj being the corresponding multiplicities. It follows from
Lemma 1.2 that there exists exactly one component, say C∗, which
1
is homologous to [E] + f∗[F], and every remaining component C∗ is
1 j
in the homology class f∗[F]. Moreover, if [C ] = [E] + f[F], then
j 0
l m f∗ = f. Applying the genus formula to C∗ we see that f∗ = 1
j=1 j j j j
fPor every j > 2.
Now recall that C∗ must pass through the p = 1 + 2f k marked
−
points x used in the definition of . On the other hand, the gener-
j h
M
icity of the path h(t) and of the points x implies that one can have at
j
most p∗ := 1+2f∗ k of the marked points on C∗, and at most one
1 1 − 1
such point on C∗ for j > 2. Altogether, this allows C∗ to pass through
j
1+2f∗ k+l 1 marked points, which is strictly less than the needed
1 − −
1+2f k unless l = 1. But this means that C∗ is irreducible and hence
−
lies in , a contradiction. Thus pr : [0,1] is proper. (cid:3)
h h h
M M →
REGULAR HOMOTOPY OF HURWITZ CURVES 7
2. Hurwitz curves
Definition 2.1. A Hurwitzcurve of degree m in the Hirzebruch surface
F is the image H¯ = f( ) F of an oriented closed real surface
N N
S ⊂ S
by a smooth map f : F E such that there exists a finite subset
N N
S → \
Z H¯ with the following properties:
⊂
(i) The restriction of f to f−1(Z) is an embedding, and for any
S\
p H¯ Z, H¯ and the fiber F of pr meet at p transversely
pr(p)
∈ \
with intersection index +1;
(ii) for each p Z there is a neighborhood U F of p such that
N
∈ ⊂
¯
H U is a complex analytic curve, and the complex orientation
∩
¯
of H U p coincides with the orientation transported from
∩ \{ }
by f;
S
(iii) the restriction of pr to H¯ is a finite map of degree m.
For any Hurwitz curve H¯ there is a unique minimal subset Z H¯
⊂
satisfying the conditions from Definition 2.1. We denote it by Z(H¯).
We say that H¯ is pr-generic if pr : Z pr(Z) is one-to-one. A fiber
|Z →
of pr is H¯-singular if it meets Z(H¯) and H¯-regular otherwise.
¯ ¯
A Hurwitz curve H has an A -singularity at p Z(H) if there is a
k
∈
neighborhood U of p and local analytic coordinates x,y in U such that
(iv) pr is given by (x,y) x;
|U 7→
(v) H¯ U is given by y2 = xk+1.
∩
An “A -singularity” is in fact a smooth point where H¯ becomes tan-
0
gent to the fiber of pr; A and A singularities are ordinary nodes and
1 2
cusps, respectively. Therefore, we will say that H¯ is cuspidal if all its
singularities are of type A with 0 6 k 6 2, and nodal if it has only
k
A and A singularities. We say that H¯ has A-singularities if all its
0 1
singularities are of type A with k > 0.
k
For our purpose we need to extend the class of admissible singular-
ities of Hurwitz curves described in Definition 2.1 (ii) by allowing the
simplest non-holomorphic one.
Definition 2.2. A negative node on a Hurwitz curve H¯ is a singular
point p Z(H¯) such that
∈
(ii−) there is a neighborhood U F of p such that H¯ U consists
N
⊂ ∩
of two smooth branches meeting transversely at p with intersec-
tion index 1, and each branch of H¯ U meets the fiber F
pr(p)
− ∩
transversely at p with intersection index +1.
8 D. AUROUX,VIK.S. KULIKOV,AND V.SHEVCHISHIN
Definition2.3. AHurwitz curve H¯ F is calledanalmost-algebraic
N
⊂
curve if H¯ coincides with an algebraic curve C over a disc D(r) P1
⊂
and with the union of m pairwise disjoint smooth sections H ,...
∞,1
H of pr over P1 D(r).
∞,m
\
Definition 2.4. Two Hurwitz curves H¯ and H¯ F (possibly with
0 1 N
⊂
negative nodes) are H-isotopic if there is a continuous isotopy φ :
t
F F , t [0,1], fiber-preserving (i.e. ψ : P1 P1 such that
N N t
→ ∈ ∃ →
pr φ = ψ pr), and smooth outside the fibers F , s Z(φ (H¯ ))
t t pr(s) t 0
◦ ◦ ∈
such that
(i) φ = Id;
0
(ii) φ (H¯ ) is a Hurwitz curve for all t [0,1];
t 0
∈
(iii) φ (H¯ ) = H¯ ;
1 0 1
(iv) φ (E ) = E for all t [0,1].
t N N
∈
In the specific case of curves with A-singularities, we can in fact
assume that φ is smooth everywhere.
t
The following theorem was proved in [Kh-Ku].
Theorem 2.5. ([Kh-Ku]) Any pr-generic Hurwitz curve H¯ F with
N
⊂
A-singularities is H-isotopic to an almost-algebraic curve. If more-
over H¯ is a symplectic surface in F , then this isotopy can be chosen
N
symplectic.
Definition 2.6. A creation of a pair of nodes along a simple curve γ is
the transformation of a Hurwitz curve H¯ = H¯t∗−τ given by a homotopy
H¯ , t [t∗ τ,t∗ +τ] and 0 < τ 1 with the following properties:
t
∈ − ≪
(1) H¯ is an isotopy outside a neighborhood U of γ;
t
(2) there exist real coordinates (x,y,u,v) in U such that the projec-
tion pr : F P1 is given by pr : (x,y,u,v) (x,y) and the
N
→ →
curve γ is given by u [ τ,τ],x = v = y = 0 ;
{ ∈ − }
(3) for every t [t∗ τ,t∗ + τ] the curve H¯ U consists of two
t
∈ − ∩
discs which are the graphs of the sections s± : (x,y) (u,v) =
t
7→
(x2 (t t∗)), y .
± − − ±
The disc(cid:0)given by D := v = y(cid:1)= 0,x [ √τ,√τ],x2 τ 6 u 6 τ x2
{ ∈ − − − }
is called the created Whitney disc.
The inverse transformation is called the cancellation of a pair of
nodes along the Whitney disc D.
REGULAR HOMOTOPY OF HURWITZ CURVES 9
(u,v) (u,v) (u,v)
γ (x,y) (x,y) (cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)D(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) (x,y)
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
p(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)p
(cid:0)(cid:1)−(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)+
t < t* t = t* t > t*
The curve γ lies in the fiber x = y = 0 and connects two points
on H¯ = H¯t∗−τ; the created nod{es p− and}p+ have opposite orienta-
tions: one is positive, the other negative. The inversion of the time t
interchanges the creation and the cancellation operations.
Definition 2.7. Two Hurwitz curves are regular homotopic if one of
them can be obtained from the other by the composition of a finite
number of H-isotopies, creations and cancellations of pairs of nodes.
The definition is motivated by the following claim. The proof is an
easy exercise.
Lemma 2.8. Let φ : F be a smooth homotopy of maps of a
t N
S →
closed oriented real surface to F with the following properties:
N
S
(1) for every t, the composition pr φ : P1 is a ramified
t
◦ S →
covering;
(2) φ is an H-isotopy in a neighborhood of all its critical points;
t
(3) φ is generic with respect to conditions (1) and (2).
t
Then φ is a regular homotopy.
t
Lemma 2.9. Let H¯ be a Hurwitz curve in F , F an H¯-regular fiber of
N
pr, γ F a simple smooth curve in F H¯ with endpoints on H¯, and U
⊂ \
any neighborhood of γ in F. Then there exists a regular homotopy φ
t
of H¯ which creates a pair of nodes along γ and is constant outside U.
Proof. It follows from the hypotheses of the lemma that γ meets H¯
transversally. So, shrinking U if needed, we can find local coordinates
(x,y,u,v) in U satisfying the condition (2) of Definition 2.6, such that
U H¯ consists oftwo discswhich arethegraphsofthemutually disjoint
∩
sections s± : (x,y) (u,v) = ( (x2 + τ), y). The result follows.
t∗−τ 7→ ± ±
(cid:3)
3. Factorization semigroups
In this section we recall the notion of braid monodromy factorization
semigroups defined in [Kh-Ku].
10 D. AUROUX,VIK.S. KULIKOV,AND V.SHEVCHISHIN
3.1. Semigroups over groups. A collection (S,B,α,λ), where S is
a semigroup, B is a group, and α : S B, λ : B Aut(S) are
→ →
homomorphisms, is called a semigroup S over a group B if for all
s ,s S
1 2
∈
s s = λ(α(s ))(s ) s = s ρ(α(s ))(s ),
1 2 1 2 1 2 2 1
· · ·
where ρ(g) = λ(g−1). If we are given two semigroups (S ,B ,α ,λ )
1 1 1 1
and (S ,B ,α ,λ ) over, respectively, groups B and B , we call a pair
2 2 2 2 1 2
h = (h ,h ) of homomorphisms h : S S and h : B B a
S B S 1 2 B 1 2
→ →
homomorphism of semigroups over groups if
(i) h α = α h ,
B ◦ S1 S2 ◦ S
(ii) λ (h (g))(h (s)) = h (λ (g)(s)) for all s S and all g B .
2 B S S 1 1 1
∈ ∈
The factorization semigroups defined below constitute, for our pur-
pose, the principal examples of semigroups over groups.
Let g be a set of elements of a group B. For each i I denote
i i∈I
{ } ∈
by O B the set of all the elements in B conjugated to g (the orbit
gi ⊂ i
of g under the action of B by inner automorphisms). Call their union
i
X = O B the full set of conjugates of g and the pair
∪i∈I gi ⊂ { i}i∈I
(B,X) an equipped group.
For any full set of conjugates X there are two natural maps r = r :
X
X X X and l = l : X X X defined by r(a,b) = b−1ab and
X
× → × →
l(a,b) = aba−1 respectively. For each pair of letters a,b X denote by
∈
R and R the relations defined in the following way:
a,b;r a,b;l
R stands for a b = b r(a,b) if b = 1 and a 1 = a otherwise;
a,b;r
· · 6 ·
R stands for a b = l(a,b) a if a = 1 and 1 b = b otherwise.
a,b;l
· · 6 ·
Now, put
= R ,R (a,b) X X, a = b if a = 1 or b = 1
a,b;r a,b;l
R { | ∈ × 6 6 6 }
and introduce a semigroup
S(B,X) = x X R
h ∈ | ∈ Ri
quotient ofthefreesemigroup ofwords withletters inX bytherelation
set . As will be seen in Section 3.2, the elements of this semigroup
R
represent factorizations of elements of the group B with factors in X,
up to Hurwitz equivalence. Introduce also the product homomorphism
α : S(B,X) B given by α (x) = x for each x X.
X X
→ ∈
Next, we define two actions λ and ρ of the group B on the set X:
x X λ(g)(x) = gxg−1 X.
∈ 7→ ∈
and ρ(g) = λ(g−1). It is easy to see that the above relation set
R
is preserved by the both actions; therefore, ρ and λ define an anti-
homomorphism ρ : B Aut(S(B,X)) (right conjugation action) and
→