La lecture à portée de main
109
pages
English
Documents
2007
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
109
pages
English
Ebook
2007
Le téléchargement nécessite un accès à la bibliothèque YouScribe Tout savoir sur nos offres
Publié par
Publié le
01 janvier 2007
Nombre de lectures
16
Langue
English
Publié par
Publié le
01 janvier 2007
Nombre de lectures
16
Langue
English
Generalized Torelli Groups
I n a u g u r a l - D i s s e r t a t i o n
zur
Erlangung des Doktorgrades der
Mathematisch-Naturwissenschaftlichen Fakult¨at
der Heinrich-Heine-Universit¨at Dus¨ seldorf
vorgelegt von
Marc Siegmund
aus Dusse¨ ldorf
8. August 2007
Diese Forschung wurde gef¨ordert durch die Deutsche
Forschungsgemeinschaft im Rahmen des Graduiertenkollegs
’Homotopie und Kohomologie’ (GRK 1150)Aus dem Mathematischen Institut
der Heinrich-Heine-Universit¨at Dus¨ seldorf
Gedruckt mit der Genehmigung der
Mathematisch-Naturwissenschaftlichen Fakult¨at der
Heinrich-Heine-Universit¨at Dusse¨ ldorf
Referent: Prof. Dr. Fritz Grunewald
Korreferent: Prof. Dr. Wilhelm Singhof
Tag der mundlic¨ hen Prufung¨ : 31.10.2007Abstract
Let F be the free group on n≥ 2 elements and Aut(F ) its group ofn n
automorphisms. A well-known representation of Aut(F ) is given byn
0 ∼ρ : Aut(F )→ Aut(F /F ) GL(n,Z),=1 n n n
0where F is the commutator subgroup of F . The kernel of ρ is calledn 1n
the classical Torelli group. In [5] Grunewald and Lubotzky construct
more representations of finite index subgroups of Aut(F ). By choosingn
a finite group G and a presentation π :F →G they obtain an integraln
linear representation ρ : Γ(G,π)→G (Z), where Γ(G,π) is a finiteG,π G,π
index subgroup of Aut(F ).n
In this thesis I study the special case G = C of this construction.2
The map ρ leads to the integral linear representationC ,π2
+σ : Γ (C ,π)→ GL(n−1,Z).−1 2
Let K denote the kernel of σ , which fits into the following exactn −1
sequence
+1→K → Γ (C ,π)→ GL(n−1,Z)→ 1. (0.1)n 2
WecallthekernelK ageneralizedTorelligroup. Thefirstmaintheoremn
of this thesis states thatK is finitely generated as a group. In the proofn
wegiveasetofgeneratorsexplicitly. Notethatthistheoremcorresponds
to the famous theorem of Nielsen and Magnus, which states that the
classical Torelli group is finitely generated.
abFurther we study the abelianized group K , which becomes by then
exaxt sequence (0.1) a GL(n−1,Z)-module. Finally we consider higher
quotients of the lower central series
K =γ (K )≥γ (K )≥γ (K )≥γ (K )≥....n 0 n 1 n 2 n 3 n
Our second main theorem states the surprising fact that for i≥ 1 the
bn,iquotientsγ (K )/γ (K )arefiniteabeliangroupsoftheform(Z/2Z)i n i+1 n
with some b ∈N .n,i 0Contents
Introduction vi
Acknowledgment xi
Notation xii
1 Presentation of Groups 1
1.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Presentations of SL(n,Z) and GL(n,Z) . . . . . . . . . . 3
1.3 Some facts about finitely presented groups . . . . . . . . 9
2 Commutator Calculus 14
3 The classical Torelli Groups 22
3.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Series of IA(F ) . . . . . . . . . . . . . . . . . . . . . . . 24n
4 Generalized Torelli Groups 28
4.1 Construction of the representation ρ . . . . . . . . . . 28G,π
4.2 The representation σ . . . . . . . . . . . . . . . . . . . 30−1
4.3 The kernel of σ . . . . . . . . . . . . . . . . . . . . . . 35−1
5 Some matrix groups 55
5.1 A modified Euclidean algorithm . . . . . . . . . . . . . . 55
5.2 Generators for the matrix groups . . . . . . . . . . . . . 57
iv6 Lower central series quotients of K 63n
6.1 Modules over SL(n,Z) and GL(n,Z) . . . . . . . . . . . 63
ab6.2 The abelianized group K . . . . . . . . . . . . . . . . . 71n
6.3 The special case n = 2 . . . . . . . . . . . . . . . . . . . 78
6.4 Higher quotients of the lower central series . . . . . . . . 80
7 Further results 85
7.1 IA(F ) as a subgroup of K . . . . . . . . . . . . . . . 85n−1 n
7.2 The relation between IA(F ) and K . . . . . . . . . . . 87n n
8 Appendix 91
Bibliography 93
vIntroduction
Let F be the free group on n≥ 2 elements and Aut(F ) its group ofn n
automorphisms. A theorem of Nielsen says that Aut(F ) is a finitelyn
presented group. A well-known representation of Aut(F ) is given byn
0 ∼ρ : Aut(F )→ Aut(F /F ) GL(n,Z),=1 n n n
0where F is the commutator subgroup of F and ρ (ϕ) is the automor-n 1n
0phism of the abelian group F /F induced by ϕ∈ Aut(F ). The kerneln nn
of ρ is called the classical Torelli group and is denoted by IA(F ).1 n
A theorem of Nielsen and Magnus ([13], [11]) says that the classical
Torelli group is finitely generated. Taking a free basis x ,...,x of F1 n n
they prove:
Theorem: The group IA(F ) is generated by the following automor-n
phisms
−1 −1 −1K :{x 7→x xx } and K :{x 7→xx x x x }ij i j i ijk i i j kj j k
(values not given are identical to the argument).
By the exactness of the sequence
1→ IA(F )→ Aut(F )→ GL(n,Z)→ 1n n
abthe abelianized group IA(F ) becomes a GL(n,Z)-module. It is a well-n
known theorem of Formanek (see [6]) that
ab n∼IA(F ) ⊗ C C ⊕V=n Z n
asaGL(n,C)-module, whereV isacertainirreducibleGL(n,C)-modulen
of dimension dim (V ) =n(n+1)(n−2)/2.
C n
In [5] Grunewald and Lubotzky construct more representations of
finite index subgroups of Aut(F ). LetG be a finite group andπ :F →n n
viG a surjective homomorphism with kernel R. Define the finite index
subgroup Γ(G,π) of Aut(F ) byn
Γ(G,π) :={ϕ∈ Aut(F )| ϕ(R) =R, ϕ induces the identity on F /R}.n n
0 ab¯Define further R := R/R = R to be the abelianization of R. Let
t denote the Z-rank of this finitely generated free abelian group. The
¯group G acts on R by conjugation. Every automorphism ϕ∈ Γ(G,π)
¯induces a linear automorphism ϕ¯ of R which is G-equivariant. Let
¯G := Aut (C⊗ R)≤ GL(t,C).G,π G Z
¯ThegroupG isthecentralizerofthegroupGactingonC⊗ RthroughG,π Z
matrices with rational entries. Define
¯ ¯G (Z) :={Φ∈G | Φ(R) =R}.G,π G,π
¯Choosing aZ-basis of R, we obtain an integral linear representation
ρ : Γ(G,π) → G (Z)G,π G,π
ϕ 7→ ϕ.¯
In the special case G ={1} this construction leads to the classical rep-
resentation ρ : Aut(F )→ GL(n,Z). Thus the kernel of ρ can be1 n G,π
considered as a natural generalization of IA(F ). Therefore it is called an
generalized Torelli group.
InmyworkIstudyanotherspecialcaseoftheconstructionbyGrune-
wald and Lubotzky. Let F (n≥ 2) be the free group generated byn
x,y ,...,y andC thecyclicgroupofordertwogeneratedbyg. More-1 n−1 2
over let π :F →C be the surjective homomorphism defined byn 2
π(x) :=g, π(y ) := 1, ..., π(y ) := 1.1 n−1
ThekernelR ofthismapis, bytheformulaofReidemeisterandSchreier,
a free group of rank 2n− 1, which means that t = 2n− 1. By the
construction above we obtain a homomorphism
¯ ∼ρ : Γ(C ,π)→ GL(R) GL(2n−1,Z).=C ,π 22
We set
+Γ (C ,π) :={ϕ∈ Γ(C ,π)| det(ρ (ϕ)) = 1}.2 2 1
viiThis is asubgroup of index two inΓ(C ,π). Animportantfeature is that2
+we are able to present a finite set of generators of Γ (C ,π) (see Chapter2
4.2). The restriction of ρ leads to the representationC ,π2
+ ¯ ∼ρ : Γ (C ,π)→ GL(R) GL(2n−1,Z).=C ,π 22
¯ ¯The Q-vector space Q⊗ R decomposes as Q⊗ R = V ⊕V , where
Z Z 1 −1
¯ ¯V , V are the±1 eigenspaces of g, respectively. Set R :=R∩V and1 −1 1 1
¯ ¯ ¯R := R∩V . It turns out that the Z-rank of R equals n and the−1 −1 1
+¯ ¯ ¯
Z-rank ofR equalsn−1. Since Γ (C ,π) leavesR andR invariant,−1 2 1 −1
we obtain representations
+ +σ : Γ (C ,π)→ GL(n,Z), σ : Γ (C ,π)→ GL(n−1,Z).1 2 −1 2
+Themapσ isequivalenttoρ restrictedtoΓ (C ,π). Incontrastthe1 1 2
representationσ is somewhat less expected and is studied in this work.−1
In Chapter 4.2 it is shown, that the map σ is surjective by analysing−1
+the images of the generators of Γ (C ,π). Let K denote the kernel of2 n
σ , which fits into the following exact sequence−1
+1→K → Γ (C ,π)→ GL(n−1,Z)→ 1.n 2
+By the exactness of this sequence, the index of K in Γ (C ,π) is infi-n 2
nite for n≥ 3 and two for n = 2. The first main theorem of this thesis
states that K is finitely generated as a group. The proof, in which then
generators are given explicitly, is provided in Chapter 4.3. As a corollary
we obtain the following theorem.
Theorem: Let n≥ 2. The group K is generated by the followingn
automorphisms:
2 2ε :{x7→xy}, ψ :{y 7→yx} ,i i i ii
( )
−1x 7→ x
α :i −1 −1y 7→ xy xi i
for 1≤ i≤ n−1 (values not given are identical to the argument). In
particular K is finitely generated as a group.n
viiiNote that this theorem corresponds to the theorem of Nielsen and Mag-
nus. The idea of the proof is the following. Starting with a finite pre-
+sentation of GL(n−1,Z) and the generator set of Γ (C ,π) we are able2
to construct a finite number of elements in K whose normal closuren
coincides with K . Then we show that the group generated by thesen
+elements is already a normal subgroup of Γ (C ,π). This means that2
K is finitely generated as a group.n
abAsaboveK becomesaGL(n−1,Z)-module. InChapter6westudyn
the structure of this module.
abProposition: Let n≥ 2. Then the group K is generated by [ε ],in
2[α ] and [ψ ] for i = 1,...,n−1. The order of [α ] is either one or two.i ii
2For n≥ 3 the order of [ψ ] is also either one or two.i
InChapter6.2weconstructforn≥ 3asurjectiveGL(n−1,Z)-equivariant
homomorphism
abΦ :V ⊕M K ,n n−1 n−1 n
where V ⊕M is a certain GL(n− 1,Z)-module with underlyingn−1 n−1
n−1 n−1 n−1abelian gr