Centralizers of Lie Algebras Associated to the Descending Central Series of Certain Poly-Free Groups

0.25 MB·English

by  Daniel C. Cohen

CENTRALIZERS OF LIE ALGEBRAS ASSOCIATED TO THE DESCENDING
CENTRAL SERIES OF CERTAIN POLY-FREE GROUPS
7 DANIELC. COHEN∗, F. R. COHEN†, ANDSTRATOS PRASSIDIS‡
0
0
2 Abstract. Poly-free groups are constructed as iterated semidirect products of free groups. The

n class of poly-free groups includestheclassical purebraid groups, fundamental groupsof fiber-type
a
hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The
J
purposeofthisarticleistocomputecentralizersofcertainnaturalLiesubalgebrasoftheLiealgebra
1
2 obtainedfromthedescendingcentralseriesofpoly-freegroupsΓincludingsomeofthegeometrically
interesting classes of groups mentioned above. The main results here extend the result in [10] for

]
R such groups. These results imply that a homomorphism f : Γ → G is faithful, essentially, if it is
G faithful when restricted to the level of Lie algebras obtained from the descending central series for
. the product FT ×Z, where FT is the “top” free group in the semidirect products of free groups
h
and Z is thecenter of Γ. The arguments use a mixtureof homological, and Lie algebraic methods
t
a applied tocertain choices of extensions. Thelimitations of these methodsare illustrated usingthe
m
“poison groups” of Formanek and Procesi [18], poly-free groups whose Lie algebras do not have
[
certain properties considered here.

2
v
0
7
4 1. Introduction
3
0 A group Γ is poly-free if there is a sequence
6
0
1 = Γ Γ Γ = Γ
/ { } 0 ≤ 1 ≤ ··· ≤ n
h
t such that Γ is a normal subgroup of Γ and the quotient Γ /Γ is isomorphic to a free group.
a i+1 i i i+1
m Examples include the Artin pure braid group, the fundamental group of the configuration space of
: n ordered points in C, as well as fundamental groups of certain orbit configuration spaces [5, 8].
v
i There are also natural subgroups of the automorphism group of a free group which are poly-free.
X
r One such example considered here is the “upper triangular” version of McCool’s subgroup of basis-
a
conjugating automorphisms [9]. Homological properties of such groups have been investigated in
[7] and [19]. Geometric properties of poly-free groups were investigated in [1] and [17].
Themain objective of this paper is to determine the centralizers of certain Lie subalgebras of the
Liealgebras whicharisefromtheclassical descendingcentralseriesofthefamilies ofdiscretegroups
2000 Mathematics Subject Classification. Primary 20E22, 20F14; Secondary 20F28, 20F36, 20F40, 32S22, 55R80.
Key words and phrases. poly-free group, descending central series, Lie algebra centralizer, McCool group, orbit
configuration space, fiber-typearrangement.
∗Partially supported byNational Security Agency grant H98230-05-1-0055.
†Partially supported by theNSF.
‡Partially supported by Canisius College SummerGrant.
1
2 D.C.COHEN,F.R.COHEN,ANDSTRATOSPRASSIDIS
given above. These Lie algebras have been used in contexts ranging from Vassiliev invariants of
pure braids [21, 22] to structures of simplicial groups and the loop space of the 2-sphere [2, 11].
Otherapplications oftheseLiealgebrasariseinthestudyofhomotopygroupsofhigherdimensional
knot spaces, see [28].
For nonempty subsets U and V of a (discrete) group G, let [U,V] denote the subgroup of G
generated by all commutators [u,v] =uvu−1v−1 of elements u U and v V. Let G be the n-th
n
∈ ∈
stage of the descending central series of G, defined inductively by G = G, and G = [G ,G] for
1 n+1 n
n 1, and let gr (G) = G /G be the n-th associated quotient. Let gr (G) = gr (G).
≥ n n n+1 ∗ Ln≥1 n
There is a bilinear homomorphism
[−,−]: grp(G)⊗Zgrq(G) −→ grp+q(G)
induced by the commutator map
c :G G G
× −→
(which is itself not in general a homomorphism).
The construction of gr (G) is a functor from the category of discrete groups to the category of
∗
Lie algebras over Z. This construction has potential applications to questions concerning repre-
sentations. Specifically, if G is residually nilpotent and a group homomorphism f out of G induces
a Lie algebra monomorphism on the corresponding Lie algebras, then f is itself a monomorphism
([11]). Based on this property, conditions which insure that a representation of G is faithful are
recorded in Corollary 1.3 below. It should be noted, however, that these methods have not yet
succeeded in producing faithful representations of the poly-free groups considered here.
1.1. Summary of the Results. Let B be a subset of the Lie algebra A. Then the centralizer of
B in A is defined by
(B) = a A : [a,b] = 0, for all b B .
A
C { ∈ ∈ }
Abbreviate (A) by (A). The main result of this paper gives centralizers of Lie subalgebras of
A
C C
gr (G) for the following groups G:
∗
I. The “upper triangular McCool groups”, subgroupsPΣ+ of the automorphism group of the
n
free group F = F[x ,...,x ] generated by automorphisms β , 1 i < j n, defined by
n 1 n i,j
≤ ≤
xk, if k = i,
β (x ) = 6
i,j k 
x−1x x , if k = j.
i j i

II. FundamentalgroupsP (n)oforbitconfiguration spaces associated to surfacegroupsGact-
G
ing freely, and properly discontinuously on the upper 1/2-plane H. In this case, gr (P (n))
∗ G
is generated by Bσ with σ G, 1 i < j n.
i,j ∈ ≤ ≤
III. Fundamental groups P(r,n) of orbit configuration spaces associated to finite cyclic groups,
of order r, acting freely on C∗, by rotation. The generators of the Lie algebra gr (P(r,n))
∗
(p)
are given by B with 1 p r, 1 i < j n, and Z for 1 k n.
i,j ≤ ≤ ≤ ≤ k ≤ ≤
CENTRALIZERS OF LIE ALGEBRAS 3
The groups Γ given by PΣ+, P (n), or P(r,n) all have the property that there is an extension
n n G
j p
(1.1) 1 F Γ Γ 1
α(n) n n−1
−−−−→ −−−−→ −−−−→ −−−−→
which satisfies the following properties.
(1.2a) Γ is a free group;
0
(1.2b) F is a free group on a countable set α(n) of cardinality at least two;
α(n)
(1.2c) the map p :Γ Γ is a split epimorphism; and
n n−1
→
(1.2d) the action of Γ on H (F ) is trivial.
n−1 ∗ α(n)
By work in [20, 14, 29], for such a group Γ , there is a split extension of Lie algebras
n
(1.3) 0 gr (F ) gr∗(j) gr (Γ ) gr∗(p) gr (Γ ) 0.
−−−−→ ∗ α(n) −−−−→ ∗ n −−−−→ ∗ n−1 −−−−→
In particular the map j :F Γ induces an injection on the level of Lie algebras
α(n) n
→
gr (j) : L[V ] = gr (F ) gr (Γ ),
∗ α(n) ∗ α(n) −→ ∗ n
where L[V ] denotes the free Lie algebra generated by the set α(n). Groups Γ which admit this
α(n) n
structure are the main focus of this paper.
Theorem 1.1. Let Γ be one of the groups described above. The centralizer of the free Lie algebra
n
L[V ] in gr (Γ ) is given as follows:
α(n) ∗ n
I. If Γ = PΣ+ is the upper triangular McCool group, then
n n
(L[V ]) = L[B ],
Cgr∗(Γn) α(n) n
where B is the image of β in gr (PΣ+), and B = n B .
i,j i,j ∗ n n j=2 1,j
P
II. If Γ = P (n) is the fundamental group of a surface group orbit configuration space, then
n G
(L[V ]) = 0.
Cgr∗(Γn) α(n)
III. If Γ = P(r,n) is the fundamental group of a cyclic group orbit configuration space, then
n
(L[V ]) = L[∆(r,n)],
Cgr∗(Γn) α(n)
where
n r
(p)
∆(r,n) = Z + B .
X k X X i,j
k=1 p=11≤i<j≤n
For a poly-free group Γ satisfying conditions (1.2), the center of the Lie algebra gr (Γ ) is
n ∗ n
determined in section 2. Namely, it is shown in Proposition 2.1 that the center of gr (Γ ) is either
∗ n
infinite cyclic or trivial. Furthermore, if the free group Γ is of rank greater than one (as is the
0
case for the groups F by (1.2b)), then both the center of gr (Γ ) and the center of Γ itself are
α(k) ∗ n n
trivial. This property need not be satisfied for an arbitrary poly-free group, as illustrated by the
direct product F F , where F is free of rank one.
1 1 1
×
4 D.C.COHEN,F.R.COHEN,ANDSTRATOSPRASSIDIS
Recall the classical adjoint representation
ad :L Der(L)
−→
of a graded Lie algebra L, where Der(L) denotes the graded Lie algebra of graded derivations of L.
The map ad is defined by the equation ad(X)(Y) = [X,Y] for X and Y in L. The center of L is
precisely the kernel of ad : L Der(L). If I is a Lie ideal of L, then the natural restriction map is
→
denoted by
ad :L Der(I).
I
| −→
LetΓ beoneofthegroupsconsideredabove. Regardgr (Γ )asagradedLiealgebrabythecon-
n ∗ n
vention that gr (Γ ) has degree 2q (as the axioms for a graded Lie algebra are not satisfied without
q n
this convention). In these cases, L[V ] is a Lie ideal. Consider the two adjoint representations:
α(n)
ad :gr (Γ ) Der(gr (Γ )) and ad : gr (Γ ) Der(L[V ]).
∗ n −→ ∗ n |L[Vα(n)] ∗ n −→ α(n)
Corollary 1.2. Let Γ be one of the above groups.
n
I. If Γ = PΣ+, then ker(ad) = ker(ad ) is the cyclic group generated by B in
n n |L[Vα(n)] n
gr (PΣ+). Thus there is a short exact sequence of Lie algebras
1 n
0 L[B ] gr (PΣ+) ad|L[Vn] Image(ad ) 0.
−−−−→ n −−−−→ ∗ n −−−−−→ |L[Vn] −−−−→
II. If Γ = P (n), then ker(ad) = ker(ad ) is trivial. Thus there is an isomorphism
n G |L[Vα(n)]
gr (P (n)) ad|L[Vn] Image(ad ).
∗ G −−−−−→ |L[Vn]
III. If Γ = P(r,n), then ker(ad) = ker(ad ) is the cyclic group generated by ∆(r,n) in
n |L[Vα(n)]
gr (P(r,n)). Thus there is a short exact sequence of Lie algebras
1
0 L[∆(r,n)] gr (P(r,n)) ad|L[Vn] Image(ad ) 0.
−−−−→ −−−−→ ∗ −−−−−→ |L[Vn] −−−−→
Recent work on theIsomorphismConjecture[15,16]has renewedinterest indetecting monomor-
phisms on discrete groups, especially when the target space is a finite dimensional linear group and
the image is a discrete subgroup. The linearity of the classical Artin braid groups, and hence
the Artin pure braid groups, was established by Bigelow [4] and Krammer [23]. Subsequently,
Digne [12] showed that Artin groups of crystallographic type are linear. This paper may be viewed
as an attempt to develop methods for detecting faithful finite dimensional representations of poly-
free groups, as illustrated by the following consequence of Theorem 1.1 and the results of [10].
Corollary 1.3. Let Γ be one of the poly-free groups PΣ+, P (n), or P(r,n), and f : Γ G a
n n G n
→
homomorphism of groups. If the morphism of Lie algebras
gr (f) : L[V ] gr (G)
∗ |L[Vα(n)] α(n) −→ ∗
is a monomorphism and
I. gr (f) :L[B ] gr (G) is a monomorphism when Γ =PΣ+;
∗ |L[Bn] n → ∗ n n
II. no further conditions when Γ = P (n);
n G
CENTRALIZERS OF LIE ALGEBRAS 5
III. gr (f) : L[∆(r,n)] gr (G) is a monomorphism when Γ = P(r,n),
∗ L[∆(r,n)] ∗ n
| →
then f is a monomorphism. In addition, the following two statements are equivalent:
(i) The map f :Γ G is one-to-one.
n
→
(ii) The maps of Lie algebras
gr (f) : L[V ] gr (f(Γ ))
∗ |L[Vα(n)] α(n) −→ ∗ n
is a monomorphism and
I. gr (f) :L[B ] gr (f(PΣ+)) is a monomorphism when Γ = PΣ+;
∗ |L[Bn] n → ∗ n n n
II. no further conditions when Γ = P (n);
n G
III. gr (f) : L[∆(r,n)] gr (f(P(r,n)) is a monomorphism when Γ = P(r,n),
∗ |L[∆(r,n)] → ∗ n
where f( ) denotes the image of f.
−
The Artin pure braid groups, and the groups P(r,n), may be realized as fundamental groups of
complements of fiber-type hyperplane arrangements. For any such arrangement, the fundamental
group of the complement is poly-free, and satisfies the conditions (1.2), see Falk and Randell [14].
Thus, there are corresponding split, short exact sequences of descending central series Lie algebras
asin(1.3). Furthermore,usingthelinearityofthepurebraidgroup[4,23]andtopologicalproperties
of fiber-type arrangements [5, 6], one can show that the fundamental group of the complement of
any fiber-type arrangement is linear, see Theorem 6.6. Consequently, many poly-free groups which
fit into exact sequences of the form (1.1) admit faithful finite dimensional linear representations.
There is, however, a dichotomy as follows. The “poison group” H of Formanek and Procesi
admits no faithful, finite dimensional linear representation [18]. While this group is poly-free, and
fits into a split exact sequence
j p
1 F H F 1,
3 2
−−−−→ −−−−→ −−−−→ −−−−→
where F is a rank k free group, information is lost upon passage to the descending central series
k
Lie algebra. Specifically, there are homomorphisms out of H which have non-trivial kernels, but
induce monomorphisms on the level of descending central series Lie algebras. The natural map
p α :H F H (H),
2 1
× −→ ×
given by the projection p : H F and the abelianization map α : H H (H), is one such
2 1
→ →
homomorphism. The map p α has non-trivial kernel but induces a monomorphism on the Lie
×
algebras. Thus, the Lie algebraic methods here fail to inform on representations of the group H.
This failure arises directly from the fact that the local coefficient system in homology is non-trivial,
that is, the action of F on H (F ) is non-trivial. For details, see Proposition 7.1.
2 ∗ 3
The above discussion suggests the following.
Question. Let Γ be a group that fits into a split exact sequence
1 F Γ G 1
−−−−→ −−−−→ −−−−→ −−−−→
6 D.C.COHEN,F.R.COHEN,ANDSTRATOSPRASSIDIS
with F a finitely generated freegroupandG agroupthat admits afinitedimensionalfaithfullinear
representation. If G acts trivially on the homology of F, does Γ then admit a finite dimensional
faithful linear representation?
2. Centers
The purpose of this section is to identify the center of the descending central series Lie algebra
gr (Γ ) in the case where Γ belongs to class of poly-free groups which satisfy the conditions (1.2).
∗ n n
Recall that such groups are given inductively by extensions
j p
1 F Γ Γ 1,
α(n) n n−1
−−−−→ −−−−→ −−−−→ −−−−→
where Γ is a free group; F is a free group of rank at least 2; the map p : Γ Γ is a split
0 α(n) n n−1
→
epimorphism; and the action of Γ on H (F ) is trivial. Denote the center of a Lie algebra g
n−1 ∗ α(n)
by (g), and write F = Γ .
α(0) 0
C
Proposition 2.1. Let Γ be a poly-free group satisfying conditions (1.2). Then the center of
n
gr (Γ ) is cyclic. Furthermore, if F is of rank at least two for every k, 0 k n, then both
∗ n α(k) ≤ ≤
the center of gr (Γ ) and the center of Γ are trivial.
∗ n n
Proof. By [14], there is a short exact sequence of Lie algebras
gr (j) gr (p)
0 gr (F ) ∗ gr (Γ ) ∗ gr (Γ ) 0,
−−−−→ ∗ α(n) −−−−→ ∗ n −−−−→ ∗ n−1 −−−−→
where gr (j) and gr (p) are the maps induced by j and p, and the first homology of Γ is free
∗ ∗ k
abelian for each k. Since Γ is free of rank at least one, the center of gr (Γ ) is either infinite cyclic
0 ∗ 0
if Γ = Z, or is trivial if the rank of Γ is at least two [10].
0 0
Assumeinductivelythatthecenterofgr (Γ )isconcentratedindegree1andiscyclic. Observe
∗ n−1
that gr (p)( (gr (Γ )), the image of the center of gr (Γ ) in gr (Γ ), is also cyclic. In addition,
∗ C ∗ n ∗ n ∗ n−1
it may be assumed inductively that the center (gr (Γ )) is trivial in case the rank of F is at
∗ n−1 α(k)
C
least two for all k, 0 k n 1.
≤ ≤ −
Choose an element ∆ in (gr (Γ )) which projects to a generator of gr (p)( (gr (Γ )). Let ∆
1 ∗ n ∗ ∗ n 2
C C
be an arbitrary element in the center (gr (Γ )). If ∆ is concentrated in degree greater than 1,
C ∗ n 2
then gr (p)(∆ ) = 0. Consequently, there is an element X in gr (F ) such that gr (j)(X) = ∆ .
∗ 2 ∗ α(n) ∗ 2
Since the center of gr (F ) is trivial whenever the cardinality of α(n) is at least two, ∆ = 0.
∗ α(n) 2
Thus it suffices to assume that ∆ is concentrated in degree 1 in case the cardinality of α(n) is at
2
least two.
It follows that there is an integer q such that q∆ and ∆ project to the same element in
1 2
(gr (Γ )). Then ∆ = q∆ ∆ is an element of (gr (Γ )), and is in the image of gr (j), say
C ∗ n−1 1− 2 C ∗ n ∗
∆ = gr (j)(δ), where δ (gr (F )). However, the center of gr (F ) is trivial whenever the
∗ ∈ C ∗ α(n) ∗ α(n)
cardinality of α(n) is at least two and so q∆ ∆ = 0. Thus the center (gr (Γ )) is cyclic, and
1− 2 C ∗ n
is trivial in the case where the rank of F is at least 2 for all k, 0 k n.
α(k)
≤ ≤
To finish, assume that the rank of F is at least 2 for all k, 0 k n. Since the center of
α(k)
≤ ≤
F is trivial for 0 k n, the center of Γ is trivial by the natural induction on n. (cid:3)
α(k) n
≤ ≤
CENTRALIZERS OF LIE ALGEBRAS 7
The previous proposition suggests the following.
Question. Let Γ be a group which fits into an exact sequence
1 F Γ G 1
−−−−→ −−−−→ −−−−→ −−−−→
where F is a free group of rank at least two and the center of gr (G) is cyclic. Is then the center
∗
of gr (Γ) cyclic? Note that no assumptions are made regarding the existence of a splitting, or the
∗
action of G on the homology of F.
3. Upper Triangular McCool Groups
Let F be the free group on the set x ,x , ,x . Let IA denote the kernel of the natural
n 1 2 n n
{ ··· }
map
Aut(F ) GL(n,Z)
n
→
induced by the map of F to its abelianization. As shown by Nielsen, and Magnus, the group IA
n n
is generated by automorphisms β , 1 i,j n, i = j, and Θ , where 1 j,s,t n and j,s,t
i,j j;[s,t]
≤ ≤ 6 ≤ ≤
are distinct, see [24]. These automorphisms are given by
β (x )= xk, if k 6= j, and Θ (x ) = xk, if k 6= j,
i,j k  j;[s,t] k 
x−1x x , if k = j, x [x ,x ], if k = j.
i j i j s t
 
Let PΣ be the group of basis-conjugating automorphisms, the subgroup of IA generated by
n n
the automorphisms β . McCool [25] showed that the group PΣ admits a presentation with these
i,j n
generators and relations
[β ,β ] for i,j,k,l distinct
 i,j k,l 
(3.1)  [βi,j,βi,k] for i,j,k distinct .
[β ,β β ] for i,j,k distinct
j,k i,j i,k
 
 
Let PΣ+ be the upper triangular McCool group, the subgroup of PΣ generated by the auto-
n n
morphisms β with i < j. The (relevant) relations (3.1) may be used to show that PΣ+ is a
i,j n
semidirect product, PΣ+ = F ⋊PΣ+ , where F is the free group on the set
n n−1 n−1 n−1
β : i= 1,2,...n 1 .
i,n
{ − }
In other words, there is a split exact sequence
1 F PΣ+ PΣ+ 1,
−−−−→ n−1 −−−−→ n −−−−→ n−1 −−−−→
see [9] (where the automorphisms β are denoted by χ(j,i)). It is readily checked that the action
i,j
of PΣ+ on H (F ) is trivial.
n−1 ∗ n−1
8 D.C.COHEN,F.R.COHEN,ANDSTRATOSPRASSIDIS
The structure of the Lie algebra gr (PΣ+) is determined in [9]. Denote the image of β in
∗ n i,j
gr (PΣ+) by B . Then gr (PΣ+) has generators B , 1 i< j n, and relations
∗ n i,j ∗ n i,j ≤ ≤
(3.2a) [B ,B ]= 0, for i,j,k,l distinct,
i,j k,l
(3.2b) [B ,B ]= 0, for i,j,k distinct,
i,j i,k
(3.2c) [B ,B +B ]= 0, for i,j,k distinct.
j,k i,j i,k
Note that the last of these is one of the classical infinitesimal braid relations. It is not, however,
the case that all of the infinitesimal braid relations are satisfied. For example, if i < j < k, the
element [B ,B +B ] need not vanish in gr (PΣ+).
i,k i,j j,k ∗ n
For 2 k n, let V = B : 1 i k 1 , and let L[V ] be the corresponding free Lie
k i,k k
≤ ≤ { ≤ ≤ − }
algebra. There is an additive isomorphism
gr (PΣ+) = L[V ].
∗ n ∼ M k
2≤k≤n
Moreover, it is clear from the relations (3.2) that the Lie algebra gr (F ) = L[V ] is a Lie ideal
∗ n−1 n
of gr (PΣ+).
∗ n
The next result is the portion of the Theorem 1.1 pertaining to Case I, where Γ = PΣ+.
n n
Theorem 3.1. The centralizer of L[V ] in gr (PΣ+) is the linear span of the element
n ∗ n
n
B = B .
n X 1,j
j=2
Proof. Proposition 2.1 implies that an element in the centralizer of L[V ] must be of weight 1. Let
n
X = a B
i,j i,j
X
1≤i<j≤n
be an element that centralizes L[V ]. Since X centralizes L[V ],
n n
[X,B ] = 0.
1,n
Relation (3.2a) implies that
n−1 n−1
a [B ,B ]+ a [B ,B ] = 0.
X 1,j 1,j 1,n X i,n i,n 1,n
j=2 i=2
Relation (3.2b) implies that each term in the first sum vanishes. Therefore,
n−1
a [B ,B ]= 0.
X i,n i,n 1,n
i=2
But the last equation takes place in the free Lie algebra L[V ]. Thus, a = 0 for 1 < i < n, and
n i,n
the element X has the form
X = a B +a B .
X i,j i,j 1,n 1,n
1≤i<j<n
CENTRALIZERS OF LIE ALGEBRAS 9
Fix j, 1< j < n. Start with the identity
[X,B ]= 0.
j,n
Relation (3.2a) implies that
j−1 n−1
a [B ,B ]+ a [B ,B ]+a [B ,B ] = 0.
X i,j i,j j,n X j,k j,k j,n 1,n 1,n j,n
i=1 k=j+1
Relation (3.2b) implies that each term in the middle sum vanishes. Thus,
j−1
a [B ,B ]+a [B ,B ] = 0.
X i,j i,j j,n 1,n 1,n j,n
i=1
Using the relation (3.2c), notice that the next relation follows
j−1
a [B ,B ]+a [B ,B ] = 0
−X i,j i,n j,n 1,n 1,n j,n
i=1
which implies that
j−1
a [B ,B ]+( a +a )[B ,B ]= 0.
−X i,j i,n j,n − 1,j 1,n 1,n j,n
i=2
Since all the commutators are linearly independent, a =0 for 1 < i< j, and a = a . Since j
i,j 1,j 1,n
was arbitrary, these relations hold for all j with 1 < j < n. The result follows. (cid:3)
4. Surface Group Orbit Configuration Spaces
Let M be a manifold without boundary, and let G be a discrete group which acts freely on M.
The orbit configuration space consists of all ordered n-tuples of points in M which lie in distinct
orbits:
ConfG(M,n) = (x ,...,x ) Mn : G x G x = if i= j .
1 n i j
{ ∈ · ∩ · ∅ 6 }
In [29], Xicot´encatl proves that, for ℓ n, projection onto the first ℓ-coordinates,
≤
p : ConfG(M,n) ConfG(M,ℓ),
G
→
is a locally trivial bundle, with fiber ConfG(M QG,n ℓ), where QG denote the union of ℓ distinct
\ ℓ − ℓ
orbits, G x ,...,G x , in M. This result generalizes the Fadell-Neuwirth theorem [13].
1 n
· ·
Let G be a discrete subgroup of PSL(2,R) acting freely and properly discontinuously on the
upper–half plane H2 by fractional linear transformations. Let ConfG(H2,n) be the corresponding
orbit configuration space and P (n) its fundamental group. Then there is a fiber bundle
G
H2 QG ConfG(H2,n) ConfG(H2,n 1).
\ n−1 → → −
This induces a split exact sequence:
1 F P (n) P (n 1) 1,
G G
−−−−→ −−−−→ −−−−→ − −−−−→
where F is the free group on QG . Notice that in this case F is an infinitely generated free group.
n−1
10 D.C.COHEN,F.R.COHEN,ANDSTRATOSPRASSIDIS
The structure of gr (P (n)) is determined in [8]. The generators are Bσ , where 1 i < j n
∗ G i,j ≤ ≤
and σ G. In this case, the infinitesimal braid relations are:
∈
[Bσ ,Bτ ]= 0, if i,j s,t = ,
i,j s,t
{ }∩{ } ∅
[Bτ ,Bτσ−1 +Bσ ]= 0, if 1 i< s < j n,
i,j s,j i,s
≤ ≤
(4.1)
[Bσ ,Bτ +Bτσ−1]= 0, if 1 i< s < j n,
i,s i,j s,j
≤ ≤
[Bτσ−1,Bτ +Bσ ]= 0, if 1 i< s < j n.
s,j i,j i,s
≤ ≤
The last equation follows from the previous two equations. Note that, for 0 < i < k < n, this
last equation implies that
(4.2) [Bτσ−1,Bτ +Bσ ] = 0.
k,n i,n i,k
Let V be the linear span of the set Bσ : 1 i < j, σ G . Then, there is a splitting
j { i,j ≤ ∈ }
n
gr (P (n)) = gr (L[V ]),
i G M i j
j=2
as abelian groups.
The next result is the portion of the Theorem 1.1 pertaining to Case II, where Γ =P (n).
n G
Theorem 4.1. The centralizer of L[V ] in gr (P (n)) is zero:
n ∗ G
(L[V ]) = 0.
Cgr∗(PG(n)) n
Proof. Proposition 2.1 implies that elements in the centralizer are of weight 1. Let
ℓ p
x= a Bσk + b Bτm
X k ik,jk X m m,n
k=1 m=1
be such an element. Since G is infinite, there exists τ G such that
∈
τσ−1 = τ , for all k = 1,2,...,ℓ, m =1,2,...,p.
k 6 m
Since x centralizes L[V ],
n
ℓ p
0 = [x,Bτ ] = a [Bσk ,Bτ ]+ b [Bτm ,Bτ ].
1,n X k ik,jk 1,n X m m,N 1,n
k=1 m=1
The infinitesimal braid relations (4.1) imply that
[Bσk ,Bτ ]= 0, if 1 = i ,
ik,jk 1,n 6 k
and that
[Bσk ,Bτ ]= [Bτσk−1,Bτ ].
1,jk 1,n − jk,n jk,n
Thus, for eack 1 i < n,
k
≤
[Bσk ,Bτ ]= [Bτσk−1,Bτ ],
ik,jk 1,n − jk,n 1,n