UTMS 2004–8

February 20, 2004

On the Kernel of the Magnus representationof the Torelli groupbyMasaaki Suzuki

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANOn the kernel of the Magnus representation ofthe Torelli group
Masaaki Suzuki
Abstract
From our previous paper, it is known that the Magnus representation of the Torelli group is not faithful. In this paper, we characterize
the kernel of its representation for a certain kind of elements.

1Introduction

The linearity of the mapping class group of a surface of genus g ≥ 2 has been
one of the well-known open problems. A group is called linear if it admits
a ﬁnite dimensional faithful representation. Recently, Korkmaz [K], Bigelow
and Budney [B-B] proved that the mapping class group of a closed surface
of genus 2 is linear. However, it still remains open for higher genera. Then
it is signiﬁcant to discuss whether some representations of the mapping classgroups are faithful and to determine the kernel.
Let Σg,1 be an oriented surface obtained from a closed surface of genus gby removing an open disk. We denote by Mg,1 the mapping class group ofΣg,1 relative to the boundary, that is the group of path components of thegroup of orientation preserving diﬀeomorphisms of Σg,1 which restrict to theidentity on the boundary. Let Ig,1 be the Torelli group of Σg,1 , namely thenormal subgroup of Mg,1 consisting of all the elements which act trivially on
the ﬁrst homology group of Σg,1 .
The Magnus representations of various subgroups of the automorphismgroup of a free group are deﬁned making use of the Fox derivation [F], see[Bir] for details. The Magnus representation for the Torelli groupr1 : Ig,1 → GL(2g; Z[H])1β2β1α2α1bb

Figure 1: Generators of Γ0 and base points b, b
was introduced in [M1], where H = H1 (Σg,1 ; Z). From our previous paper[S1], the representation r1 is not faithful for g ≥ 2. Thus it makes sense tostudy the kernel of r1 . In this paper, we characterize the kernel of r1 forthe commutator of two BSCC maps, where the Dehn twist along a bounding
simple closed curve is called BSCC map. The following is one of the main
result of this paper.
Corollary 4.4 The commutator of two BSCC maps ϕ1 , ϕ2 belongs to thekernel of r1 if and only if the characteristic polynomial of the Magnus matrixof the product ϕ1 ϕ2 is trivial. Here the Magnus matrix means the image of
r1 for a mapping class.
In Section 2, we will recall the deﬁnitions of the Magnus representationof the mapping class group and the Torelli group.
In Section 3, we will give a certain pairing for two curves on Σg,1 andshow the relationship with the pairing and the kernel of r1 .
In Section 4, we will introduce another pairing for two curves on Σg,1 inorder to obtain additional information of the kernel of r1 .

2

Definition of the Magnus representation of
the Torelli group

In this section, we recall the deﬁnitions of the Magnus representation for themapping class group and the Torelli group from [M1], [S1] and [S4].
Let Z[Γ0 ] be the integral group ring of Γ0 = π1 (Σg,1 , b). We ﬁx a systemof generators α1 , . . . , αg , β1 , . . . , βg of the free group Γ0 as shown in Figure
1. Let us simply write γ1 , . . . , γ2g for them.

2Definition 2.1 We call the mappingr : Mg,1 −→ GL(2g; Z[Γ 0 ])∂ϕ(γj )ϕ−→
∂γii,jthe Magnus representation for the mapping class group, where ∂γ∂ i : Z[Γ0 ] →Z[Γ0 ] is the Fox derivation and ¯ : Z[Γ0 ] → Z[Γ0 ] is the antiautomorphism
induced by the mapping γ → γ −1 .
This mapping is not a homomorphism but a crossed homomorphism.
Proposition 2.2 (Morita [M1]) For any two elements ϕ, ψ ∈ Mg,1 , wehaver(ϕψ) = r(ϕ) · ϕ r(ψ)
where ϕ r(ψ) denotes the matrix obtained from r(ψ) by applying the automorphism ϕ : Z[Γ0 ] → Z[Γ0 ] on each entry.
It follows that if this mapping r is restricted to the Torelli group Ig,1 andare reduced the coeﬃcients to Z[H], then we obtain the following genuine
representation:
r1 : Ig,1 −→ GL(2g; Z[H]).
Here the reduction is induced by the abelianization a : Γ0 → H and r1denotes the composition r of the mapping r by the abelianization a. We
call r1 the Magnus representation of the Torelli group.
→We have another deﬁnition of this representation (see [S4]). Let p : ΣΣg,1 be the universal abelian covering, that is, the regular covering corresponding to the abelianization. An arbitrary element of the Torelli group p−1 (b); Z) as a free Z[H]-module of rank
induces an automorphism of H1 (Σ,2g. Therefore we get the following representation:
r1 : Ig,1 −→ GL(2g; Z[H]).

3

3

A higher intersection number of two loops
and the kernel of r1

The non-triviality of the kernel of r1 for g ≥ 2 is proved in [S1]. Moreover, itis proved in [S2] that none of the terms of the lower central series of Ig,1 iscontained in the kernel. Then it is interesting to characterize and determine
the kernel.
First, we deﬁne a pairing of two loops on Σg,1 . This pairing is useful togive information about the kernel of r1 . Choose base points b and b on ∂Σg,1
as depicted in Figure 1. Fix a point b̂ which is a lift of b to the universal The point b̂ is determined as follows, which is a lift of b .
abelian covering Σ.
We denote by bb the path on ∂Σg,1 from b to b with the orientation which be the lift of bb to Σ starting at b̂. Thenis opposite to that of Σg,1 . Let bb .
we set b̂ for the endpoint of bb
Definition 3.1 Let c1 , c2 be two oriented loops on Σg,1 based at b, b respectively. We deﬁne(hĉ1 , ĉ2 ) h.
c1 , c2 H =h∈H starting at b̂, ĉ2 is the lift of c2 to Σ starting
Here ĉ1 is the lift of c1 to Σat b̂ and (·, ·) denotes the algebraic intersection number of two arcs. Wewrite hĉ1 for the curve which is acted on ĉ1 by an element h of the coveringtransformation group H.
Suppose that c1 and c2 are bounding simple closed curves on Σg,1 , wherebounding means 0-homologous. If we regard c1 , c2 as oriented loops based at b,b respectively, then we can compute the pairing c1 , c2 H up to multiplicationby ±1 and by an element of H. That is to say, the pairing c1 , c2 H depends
on how c1 , c2 are represented as loops. However, whether c1 , c2 H is zero or
not does not depend on the choices, and we will use this fact.
Proposition 3.2 Suppose that c1 and c2 are two bounding simple closed
curves on Σg,1 , and ϕ1 and ϕ2 the Dehn twists along c1 and c2 respectively.
If c1 , c2 H = 0, then [ϕ1 , ϕ2 ] ∈ ker r1 .

4

c1
c1 t1c2 c2
c1 c2 t2
Figure 2: geometric intersection number 2Proof. We denote by ϕ∗ the automorphism of the ﬁrst homology group p−1 (b); Z) induced by a diﬀeomorphism ϕ of Σg,1 representing an elH1 (Σ, respectively. Then [ĉ1 ], [ĉ2 ]ement of Mg,1 . Let ĉ1 , ĉ2 be lifts of c1 , c2 to Σ
 p−1 (b); Z). Since c1 , c2 H = 0, the intersection number
belong to H1 (Σ,(ĉ1 , ĉ2 ) equals zero. For a loop c based at b, we denote by ĉ a lift of c to Σ.
 p−1 (b); Z) andThen we have an element [ĉ] of H1 (Σ,ϕi ∗ ([ĉ]) = [ĉ] + (ĉi , ĉ)[ĉi ]i = 1, 2.

Then we obtain2 ∗ ([ĉ]) = ϕ
1 ∗ ([ĉ] + (ĉ2 , ĉ)[ĉ2 ])ϕ
1 ∗ ◦ ϕ= [ĉ] + (ĉ1 , ĉ)[ĉ1 ] + (ĉ2 , ĉ)[ĉ2 ]1 ∗ ([ĉ]).
= ϕ2 ∗ ◦ ϕ2 ∗ and this completes the proof.
It follows that ϕ1 ∗ commutes with ϕCorollary 3.3 Suppose that c1 and c2 are two bounding simple closed curves.
If the geometric intersection number of c1 and c2 is two, then [ϕ1 , ϕ2 ] ∈ ker r1 .
Proof. Let t1 , t2 be the intersection points. Also, let ci  be the subarcs of
ci from t1 to t2 , ci  from t2 to t1 , see Figure 2. The number of the terms ofc1 , c2 H is two. Each value of the terms is decided by the value at t1 and t2
respectively. We consider loops c1  c2  , c1  c2  −1 , c2  c1  and c2  c1  −1 , where c−1
is the same arc as c with the opposite orientation. All of these are bounding
simple closed curves. It follows that the value at t1 is −1 times that of t2 .
Then c1 , c2 H = 0. By Proposition 3.2, this completes the proof.

5

4

Another pairing of bounding simple closed
curves and the kernel of r1

We deﬁne another pairing for two bounding simple closed curves:
c1 , c2  = −c1 , c2 H ·c2 , c1 H .
The pairing ·, ·H depends on the way to assigning orientations and attachingbasepoints to two bounding simple closed curves. However, the way does nothave an eﬀect on the pairing ·, ·. That is, we obtain the following lemma.
Lemma 4.1 Let c1 , c2 be two bounding simple closed curves on Σg,1 . Then
we have
1. c1 , c2  = c2 , c1 
2. γc1 γ −1 , c2  = c1 , c2 
3. c1 −1 , c2  = c1 , c2 
where γ is a loop based at b and c1 −1 is the same loop as c1 with the opposite
orientation.
We recall the following before proving Lemma 4.1.
Theorem 4.2 (Morita [M1]) There exists a matrix J such that for any
element f ∈ Mg,1 the following equality holds:t r(f ) J
 r(f ) = f J.

This means that the Magnus representation of the mapping class groupis symplectic in a sense. The explicit expression of J can be found in [M1]and [S4] and is not included into this paper.


∂c
∂c
t−→In this section, c denotes a ∂γ,.
.
.
,a.
∂γ2g
1Proof.
1. It is obvious from the deﬁnition of the pairing ·, ·.

6

2. We can consider γ as an element of Γ0 naturally. Because
 −1 
∂γc1 γ −1∂γ∂γ∂c1a= a+ a(γ)a+ a(γ)a(c1 )a
∂γi
∂γi
∂γi
∂γi

∂c1= a(γ)a,∂γithen we get−−−−→→c1 .
γc1 γ −1 = a(γ)−→→
 = J1 ,c2 J 1 −c1 , where a(J)By [S4, Lemma 4.4], we have c1 , c2 H = −t −therefore→→→→γc1γ −1 , c2  = −t −c1 J 1 −c2 J1 a(γ) −c1 t a(γ)−
c2
→
→
→
→
c2 J 1 −c1 t −
c1 J 1 −c2= −t −= c1 , c2 .
3. Sincea∂c1 −1∂γi−1= a(c1 )a

∂c1
∂γi




= −a∂c1∂γi,we deduce this lemma.
The relation between the pairing ·, · and the Magnus representation
r1 of the Torelli group can be expressed as the following formula.
Theorem 4.3 Suppose that c1 and c2 are two bounding simple closed curves
on Σg,1 . Then we obtainc1, c2  = tr(I2g − r1 (ϕ1 ϕ2 )) = 2g − tr(r1 (ϕ1 ϕ2 ))where ϕ1 , ϕ2 are the Dehn twists along c1 , c2 respectively.
Proof. Any bounding simple closed curve can be written as f (dk ) for acertain element f ∈ Mg,1 and for a bounding simple closed curve dk whichis shown in Figure 3. First, we will prove the statement in the case c1 =f (di), c2 = dj . That is, we will consider the case ϕ1 = f ψi f −1 , ϕ2 = ψj ,7

1kk+1gdk
Figure 3: bounding simple closed curve
where ψk is the Dehn twist along dk . By Lemma 4.1, we can assume that c1and c2 have expressions asc1 = f ([βi , αi ] · · · [β1 , α1 ]),c2 = [βj , αj ] · · · [β1 , α1 ].

We see from [S3] that
r1 (ψk ) = I2g + ak bk .

(4.1)

Herebkt

0 · · · 0 1 − x̄1 · · · 1 − x̄k 0 · · · 0 )g − k times
g − k times= (1 − x̄1 · · · 1 − x̄k 0 · · · 0 1 − ȳ1 · · · 1 − ȳk 0 · · · 0 ),g − k times
g − k times

ak =(ȳ1 − 1 · · · ȳk − 1and xi , yi are the homology classes of αi , βi respectively. Note that tr(ak bk ) =bk ak = 0. We denote by r the composition of the mapping r by the abelianization a : Z[Γ0 ] → Z[H]. If we consider elements of the Torelli group, wewrite r1 for r as before. By the abelianization, Theorem 4.2 can be statedast r (f ) J

1 r (f ) =

f

J1 .

(4.2)

The following equalities can be checked easily:bk J1 −1 = t ak ,8ta J = b .
k 1k(4.3)

→We will compute −c1 by an explicit calculation. Since∂c1a∂γli∂f ([βk , αk ])
=a
∂γl
k=1

i

∂f (αk )∂f (βk )=a+ a(f (βk )) · a
∂γ
∂γll
k=1



∂f (βk −1 )∂f (αk −1 )+a(f (βk )) · a(f (αk )) · a+ a(f (αk )) · a
∂γl
∂γl


i

∂f (αk )∂f (βk )=a+ f (yk ) · a
∂γl
∂γl
k=1



∂f (βk )∂f (αk )−f (xk ) · a−a
∂γl
∂γl

i

∂f (αk )∂f (βk )(f (yk ) − 1) · a+ (1 − f (xk )) · a,
=
∂γ
∂γll
k=1
we obtain−→c1 = r (f ) · f ai .

(4.4)

→
Similarly, −c2 = aj . Thereforetr(I2g − r1 (ϕ1 ϕ2 ))= tr(I2g − r (f ) · f r1 (ψi ) · r (f )−1 · r1 (ψj ))= tr(I2g − r (f ) · (I2g + f ai f bi ) · r (f )−1 · (I2g + aj bj ))Because (4.1)
f−1f−1ff= tr(−r (f ) · ai bi · r (f ) − aj bj − r (f ) · ai bi · r (f ) · aj bj )= −tr(r (f ) · f ai f bi · r (f )−1 · aj bj )= −tr(r (f ) · f ai f bi · f J1 −1 · t r (f ) · J1 · aj bj )= −tr(r (f ) ·=
=
=
=

f

ai tf ai · t r (f ) · J1 · aj t aj · J1 )−tf ai · t r (f ) · J1 · aj · tr(r (f ) · f ai t aj · J1 )→→c1 J 1 −c2 · t aj J1 r (f ) · f ai−t −Because (4.4)→→→→
c1 J 1 −c2 · t −c2 J 1 −c1−t −
c2 , c1  = c1, c2 .
9Because (4.2)
Because (4.3)

Next, we consider the general case ϕ1 ϕ2 = g f ψi f −1 ψj g −1 for g ∈ Mg,1 . The
pairing ·, · is Mg,1 -equivariant by [S4, Lemma 4.3], that is,g(c1), g(c2) = g(c1, c2 ).
Moreover, we see from [S3, Proposition 3.2] thattr(r1 (gϕ1ϕ2 g −1 )) = g(tr(r1 (ϕ1 ϕ2 ))).
This means that tr(r1 (·)) is also Mg,1 -equivariant. Therefore this completes
the proof.
The Dehn twist along a bounding simple closed curve is called a BSCC
map. From our previous paper [S3], it is known that any BSCC map ϕ does
not lie in the kernel of r1 , and the characteristic polynomial of the Magnus
matrix of ϕ is trivial:det(λI2g − r1 (ϕ)) = (λ − 1)2g .
It follows that Kg,1 is not contained in the kernel of r1 , where Kg,1 denotes thesubgroup generated by the BSCC maps. We remark that the characteristicpolynomial of the Magnus matrix on Kg,1 is not always trivial (see [S4] fordetails).
Theorem 4.3 gives a characterization of the kernel of r1 for the commutator of two BSCC maps.
Corollary 4.4 The commutator of two BSCC maps ϕ1 , ϕ2 belongs to thekernel of r1 if and only if the characteristic polynomial of the Magnus matrixof the product ϕ1 ϕ2 is trivial. Here the Magnus matrix means the image ofr1 for a mapping class.
Proof. In general, if the characteristic polynomials of two matrices A, B
are trivial and A commutes with B, then the characteristic polynomial ofAB is also trivial.
Suppose that the commutator of two BSCC maps ϕ1 , ϕ2 belongs to thekernel of r1 , that is, r1 (ϕ1 ) commutes with r1 (ϕ2 ). Because the characteristic
polynomial of the Magnus matrix for any BSCC map is trivial, we getdet(λI2g − r1 (ϕ1 ϕ2 )) = (λ − 1)2g .

10

Conversely, suppose that the characteristic polynomial is trivial. Then
we have−tr(r1 (ϕ1 ϕ2 )) = −2g.
By Theorem 4.3, we conclude that c1, c2  = 0. This means c1 , c2 H = 0 or
c2 , c1 H = 0, because Z[H] is an integral domain. In virtue of Proposition
3.2, c1 , c2 H = 0 gives [ϕ1 , ϕ2 ] ∈ ker r1 and c2 , c1 H = 0 gives [ϕ2 , ϕ1 ] ∈ker r1 . This completes the proof.

5Acknowledgements

The author would like to express his gratitude to Prof. Shigeyuki Morita for
helpful suggestions and encouragements. He also would like to thank Prof.
Yasushi Kasahara and Prof. Takayuki Morifuji for valuable discussions andadvices.

References[Big] S. Bigelow, The Burau representation is not faithful for n = 5, Geom.
Topol. Vol.3 (1999) 397-404[B-B] S. Bigelow and R. Budney, The mapping class group of a genus twosurface is linear, Alg. Geom. Topol. Vol.1 (2001) 699-708[Bir] J. Birman, Braids, Links and Mapping Class Groups, Annals of Math.
Studies No. 82, Princeton University Press, 1975
[F] R. H. Fox, Free Diﬀerential Calculus I, Annals of Math. 57 (1953), 547560[J] D. Johnson, A survey of the Torelli group, Contemporary Math. 20(1983) 165-179[K] M. Korkmaz, On the linearity of certain mapping class groups, Turkish
J. Math. 24 (2000) 367-371[M1] S. Morita, Abelian quotients of subgroups of the mapping class group ofsurfaces, Duke Math. J. 70 (1993), 699-726
11[M2] S. Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, Geom. Topol. Monogr. Vol.2: Proceedings of theKirbyfest (1999) 349-406[S1] M. Suzuki, The Magnus representation of the Torelli group Ig,1 is notfaithful for g ≥ 2, Proc. Amer. Math. Soc. 130 (2002) 909-914[S2] M. Suzuki, Irreducible decomposition of the Magnus representation ofthe Torelli group, Bull. Austral. Math. Soc. 67 (2003) 1-14
[S3] M. Suzuki, A class function on the Torelli group, Kodai Math. J. 26(2003) 304-316[S4] M. Suzuki, A geometric interpretation of the Magnus representation ofthe mapping class group, preprint

12

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