Generalized Torelli groups [Elektronische Ressource] / vorgelegt von Marc Siegmund

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• Mathematics

Informations

• Publié par: heinrich-heine-universitat_dusseldorf
• Publié le: 01 janvier 2007
• Nombre de lectures: 16
• Langue: English

Extrait

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