Table Of ContentL2-TORSION INVARIANTS AND THE MAGNUS
REPRESENTATION OF THE MAPPING CLASS GROUP
8
0 TERUAKIKITANOANDTAKAYUKIMORIFUJI
0
2 Dedicated toProfessor Shigeyuki Morita on his 60th birthday
n
a Abstract. Inthispaper, westudyaseriesofL2-torsioninvariantsfromthe
J viewpointofthemappingclassgroupofasurface. Weestablishsomevanishing
9 theorems for them. Moreover we explicitly calculate the first two invariants
2 andcomparethemwithhyperbolicvolumes.
]
T
G
1. Magnus representation
.
h
Let Σ be a compact oriented smooth surface of genus g with a boundary
t g,1
a ∂Σ ∼= S1. In this paper, we always assume that g ≥ 1. We take and fix a base
g,1
m
point ∗ ∈ ∂Σ of Σ . Let M be the mapping class group of Σ , namely,
g,1 g,1 g,1 g,1
[ the group of all isotopy classes of orientation preserving diffeomorphisms of Σ
g,1
relativeto the boundary. We denoteπ (Σ ,∗)by Γ, which is a free groupofrank
1 1 g,1
2g, and fix a generating system Γ = hx ,...,x i. Let ZΓ be the group ring of
v 1 2g
9 Γ over Z. We write ϕ ∈ Aut(Γ) to the automorphism induced from ϕ ∈ Mg,1.
2 Thefollowingresult,us∗uallycalledthe Dehn-Nielsen-Baertheorem,isclassicaland
4 fundamental to study the mapping class groupM by using combinatorialgroup
g,1
4
theories (see [9] Section 2.9).
.
1
Proposition1.1(Zieschang[27]). Theaboveinducedhomomorphism M ∋ϕ7→
0 g,1
8 ϕ ∈Aut(Γ) is injective.
∗
0
Asacorollary,weseethatϕcanbedeterminedbythewordsϕ (x ),...,ϕ (x )∈
v: Γ. Sincethefundamentalformulaγ =1+ 2g (∂γ/∂x )(x −1)h∗old1sinZΓ∗for2agny
i=1 i i
Xi γ ∈ Γ, the word ϕ (xj) is determined by {∂ϕ (xj)/∂xi}. Here ∂/∂xi : ZΓ → ZΓ
∗ P ∗
denotes Fox’s free differential. See [1] Section 3.1 for a systematic treatment of
r
a the subject. The Magnus representation of the mapping class group is defined as
follows.
Definition 1.2. The Magnus representationof M is defined by the assignment
g,1
∂ϕ (x )
r :Mg,1 ∋ϕ7→ ∗ j ∈GL(2g,ZΓ),
∂x
i !
ij
where λ g = λ g 1 for any element λ g ∈ZΓ.
g g g g − g g
RemaPrk 1.3. ByPthe expression γ = 1+ Pi(∂γ/∂xi)(xi−1), it follows that r is
injective. However,itisnotagrouphomomorphism,justacrossedhomomorphism.
P
According to the practice, we call it simply the Magnus representation of M .
g,1
This research was partially supported by Grant-in-Aid for Scientific Research (No.17540064
andNo.17740032), theMinistryofEducation, Culture,Sports,ScienceandTechnology, Japan.
1
2 T.KITANOANDT.MORIFUJI
Now for a matrix B ∈M(n,C), let us recall that its characteristic polynomial
det(tI−B)
is one of the fundamental tools in the linear algebra. Here I denotes the identity
matrix of degree n. If we can define a characteristic polynomial of r(ϕ), it may be
useful tool to study the mapping class group. In order to define it for a Magnus
matrix r(ϕ), we need to clarify the following two points.
(1) What is the determinant over a non-commutative group ring?
(2) What is the meaning of a variable “t” in the group?
As an answer to these problems, we can formulate that
• the variable t lives in the fundamental group of the mapping torus of ϕ,
• a characteristic polynomial “det”(tI −r(ϕ)) with respect to the Fuglede-
Kadison determinant.
In the later sections, we explain that the characteristic polynomial of r(ϕ) is
defined as a real number and it essentially gives the L2-torsion and the hyperbolic
volume of the mapping torus of ϕ. Moreover taking the lower central series of the
surface group Γ, we obtain a family of Magnus representations, so that we can
introduce a sequence of L2-torsion invariants as an approximate sequence of the
hyperbolic volume.
This paper is organized as follows. In the next section, we briefly recall the
definition of the Fuglede-Kadison determinant. In Section 3, we summarize some
properties of the L2-torsion of 3-manifolds and explain a relation to the Magnus
representation. We introduce a sequence of L2-torsion invariants for a surface
bundle over the circle in Section 4 and give some formulas for them in Section 5.
In the last section, we discuss some vanishing theorems for L2-torsion invariants.
2. Fuglede-Kadison determinant
In this section, we review the combinatorial definition of the Fuglede-Kadison
determinant over a non-commutative group ring and its basic properties (see [19]
for details).
The idea to define a determinant over a group ring comes from the following
observation. That is, for a matrix B ∈ GL(n,C) with the (non-zero) eigenvalues
λ ,...,λ , we can formally calculate
1 n
n n
log|detB|2 =log λ λ = logλ λ
i i i i
i=1 i=1
Y X
n ∞ (−1)p+1 p
= λ λ −1
i i
p
!
i=1 p=1
X X (cid:0) (cid:1)
n
∞ 1 p
=− 1−λ λ
i i
p
!
p=1 i=1
X X (cid:0) (cid:1)
=− ∞ 1tr((I−BB )p)
∗
p
p=1
X
L2-TORSION INVARIANTS AND THE MAGNUS REPRESENTATION 3
by the power series expansion of log, where B is the adjoint matrix of B. More
∗
precisely, if we take a sufficiently large constant K >0, we obtain
|detB|=Knexp −1 ∞ 1tr I −K−2BB∗ p ∈R>0.
2 p
!
Xp=1 (cid:16)(cid:0) (cid:1) (cid:17)
Thus if we can define a certain “trace” over a group ring, we get a determinant by
using this formula.
Let π be a discrete group and Cπ denote its group ring over C. For an element
g πλgg ∈Cπ, we define the Cπ-trace trCπ :Cπ →C by
∈
P
trCπ λgg =λe ∈C,
!
g π
X∈
where e is the unit element in π. For an n×n-matrix B = (b ) ∈ M(n,Cπ), we
ij
extend the definition of Cπ-trace by means of
n
trCπ(B)= trCπ(bii).
i=1
X
Next let us recall the definition of the L2-Betti number of an n × m-matrix
B ∈M(n,m,Cπ). We consider the bounded π-equivariant operator
R :⊕n l2(π)→⊕m l2(π)
B i=1 i=1
defined by the natural right action of B. Here l2(π) is the complex Hilbert space
of the formal sums λ g which are square summable. We fix a positive real
g π g
number K so that K ≥∈||R || holds, where ||R || is the operator norm of R .
B B B
P ∞ ∞
Definition 2.1. The L2-Betti number of a matrix B ∈M(n,m,Cπ) is defined by
b(B)= lim trCπ I−K−2BB∗ p ∈R 0,
p ≥
→∞ (cid:16)(cid:0) (cid:1) (cid:17)
where B =(b ) and λ g = λ g 1 for each entry.
∗ ji g g −
Roughlyspeaking,tPheL2-BetPtinumberb(B)measuresthesizeofthekernelofa
matrix B. Hereafter we assume b(B)=0. Then, for a matrix with coefficients in a
non-commutative group ring, we can introduce the desired determinant as follows.
Definition 2.2. The Fuglede-Kadison determinant of a matrix B ∈ M(n,m,Cπ)
is defined by
detCπ(B)=Knexp −1 ∞ 1trCπ I−K−2BB∗ p ∈R>0,
2 p
!
Xp=1 (cid:16)(cid:0) (cid:1) (cid:17)
ifthe infinite sumofnon-negativerealnumbers inthe aboveexponentialconverges
to a real number.
Remark 2.3. ItisshownthattheL2-Bettinumberb(B)andtheFuglede-Kadison
determinantdetCπ(B)areindependent ofthe choiceofthe constantK (see [16]for
example).
Hereweconsidertheconditionoftheconvergence. ForanymatrixB ∈M(n,C),
the condition
lim tr I−K 2BB p =0
− ∗
p
→∞ (cid:16)(cid:0) (cid:1) (cid:17)
4 T.KITANOANDT.MORIFUJI
implies that B has no zero eigenvalues, and then |detB| converges. In the case
of group rings, if detCπ(B) converges, then b(B) = 0. But it is not a sufficient
condition,sothatweneedadditionalone. ItisaproblemtodecidewhendetCπ(B)
converges. Undertheassumptionthatb(B)=0,suchasufficientconditionisgiven
by the positivity of the Novikov-Shubin invariant α(B). Then the convergence of
the infinite sum in the Fuglede-Kadison determinant is guaranteed. The Novikov-
Shubin invariant of an operator R measures how concentrated the spectrum of
B
R R is. However, in general, it is hard to check the positivity of the Novikov-
B∗ B
Shubin invariant.
To avoid the difficulty, we need to consider the determinant class condition for
groups (see [19], [24] for details). A group π is of det ≥ 1-class if for any B ∈
M(n,m,Zπ) the Fuglede-Kadison determinant of B satisfies detCπ(B)≥1. There
are no known examples of groups which are not of det≥ 1-class. Further recently
it was proved that there is a certain large class G of groups for which they are of
det ≥ 1-class. It includes amenable groups and countable residually finite groups.
If we can see that π belongs to G, namely it is of det ≥ 1-class, the convergence
of the Fuglede-Kadison determinant is guaranteed when the L2-Betti number is
vanishing. See [18], [19], [24] for definitions and properties of these subjects.
3. L2-torsion of 3-manifolds
In this section, we quickly recall the definition of the L2-torsion of 3-manifolds.
Itis anL2-analogueofthe Reidemeisterandthe Ray-Singertorsionandessentially
gives Gromov’s simplicial volume under certain general conditions [2], [3], [4], [8],
[14],[15],[20],[21],[22]. See[19]anditsreferencesforhistoricalbackground,related
works and so on.
Let M be a compact connected orientable 3-manifold. We fix a CW-complex
structure on M. We may assume that the action of π M on the universal cover-
1
ing M is cellular (if necessary, we have only to take a subdivision of the original
structure). We consider the Cπ M-chain complex
1
f
0−→C (M,C)−∂→3 C (M,C)−∂→2 C (M,C)−∂→1 C (M,C)−→0
3 2 1 0
of M. Since the boundary operator ∂ is a matrix with coefficients in Cπ M, if we
f f i f f 1
take the adjoint operator ∂ : C (M,C) → C (M,C) as in the previous section,
i∗ i 1 i
wefcan define the ith (combinato−rial) Laplace operator ∆ :C (M,C)→C (M,C)
i i i
by f f
∆ =∂ ◦∂ +∂ ◦∂ . f f
i i+1 i∗+1 i∗ i
Let us suppose that all the L2-Betti numbers b(∆ ) vanish and the fundamen-
i
tal group π M is of det ≥ 1-class. Thereby as a generalization of the classical
1
Reidemeister torsion, the L2-torsion τ(M) is defined by
Definition 3.1.
3
τ(M)= detCπ1M(∆i)(−1)i+1i ∈R>0.
i=0
Y
AsforthepositivityofNovikov-Shubininvariantsα(∆ )fortheLaplaceoperator
i
∆ , it is known that α(∆ ) > 0 holds under some general assumptions (see [15]).
i i
For example, if a compact connected orientable 3-manifold M satisfies
(1) π M is infinite,
1
L2-TORSION INVARIANTS AND THE MAGNUS REPRESENTATION 5
(2) M is an irreducible 3-manifold or S1×S2 or RP3♯RP3,
(3) if ∂M 6=φ, it consists of tori,
(4) if ∂M = φ, M is finitely covered by a 3-manifold which is a hyperbolic,
Seifert or Haken 3-manifold,
then b(∆ ) = 0 and α(∆ ) > 0 for each i. Therefore, we see that the L2-torsion
i i
τ(M) is also well-defined in view of these conditions.
Remark 3.2. The above condition (4) is automatically satisfied by Perelman’s
proof of Thurston’s Geometrization Conjecture.
As a notable property of the L2-torsion,it is known that logτ(M) can be inter-
pretedasGromov’ssimplicialvolume||M||andhyperbolicvolumevol(M)(see[7])
of M. See [21] for the heart of the proof.
Theorem 3.3. Let M be a compact connected orientable irreducible 3-manifold
with an infinite fundamental group such that ∂M is empty or a disjoint union of
incompressible tori. Then it holds that
logτ(M)=C||M||,
where C is the universal constant not depending on M. In particular, if M is a
hyperbolic 3-manifold, we obtain
1
logτ(M)=− vol(M).
3π
Next we review Lu¨ck’s formula for the L2-torsion of 3-manifolds ([16] Theorem
2.4). From this formula, we see that logτ is a characteristic polynomial of the
Magnus representation of the mapping class group.
Theorem 3.4. Let M be as in the above theorem. We suppose that ∂M is non-
empty and π M has a deficiency one presentation
1
hs ,...,s | r ,...,r i.
1 n+1 1 n
Put A to be the n × n-matrix with entries in Zπ M obtained from the matrix
1
(∂r /∂s ) by deleting one of the columns. Then the logarithm of the L2-torsion
i j
of M is given by
logτ(M)=−2logdetCπ1M(A)
=−2nlogK+ ∞ p1trCπ1M I−K−2AA∗ p ,
Xp=1 (cid:16)(cid:0) (cid:1) (cid:17)
where K is a constant satisfying K ≥||R || .
A
∞
To see a relation between the Magnus representation and the L2-torsion, we
describe the above Lu¨ck’s formula for a surface bundle over the circle.
For an orientation preserving diffeomorphism ϕ of Σ , we form the mapping
g,1
torus M by taking the product Σ ×[0,1] and gluing Σ ×{0} and Σ ×{1}
ϕ g,1 g,1 g,1
via ϕ. This gives a surface bundle over S1. Its diffeomorphism type is determined
bythemonodromymapϕ,andconverselythemonodromymapϕisdeterminedby
agivensurfacebundle upto conjugacyandisotopy. Here anisotopyfixessetwisely
the points on the boundary ∂Σ . We take a deficiency one presentation of the
g,1
fundamental group π =π (M ,∗),
1 ϕ
π = x1,...,x2g,t | ri :txit−1 =ϕ (xi), 1≤i≤2g ,
∗
(cid:10) (cid:11)
6 T.KITANOANDT.MORIFUJI
where the base point ∗ of π and Γ = π (Σ ,∗) is the same one on the fiber
1 g,1
Σ ×{0}⊂M and ϕ :Γ→Γ is the automorphism induced by ϕ:Σ →Σ .
g,1 ϕ g,1 g,1
Itshouldbe notedthat∗π isisomorphictothesemi-directproductofΓandπ S1 ∼=
1
Z=hti.
Applying the free differentialcalculus to the relationsr (1≤i≤2g),we obtain
i
the Alexander matrix
∂r
A= i ∈M(2g,Zπ).
∂x
(cid:18) j(cid:19)
Then Lu¨ck’s formula for a surface bundle over the circle is given by
logτ(Mϕ)=−2logdetCπ(A)
=−4glogK+ ∞ 1trCπ I −K−2AA∗ p ,
p
Xp=1 (cid:16)(cid:0) (cid:1) (cid:17)
where K is a constant satisfying K ≥||R || .
A
ThisformulaenablesustointerprettheL∞2-torsionlogτ ofasurfacebundleover
the circle as the characteristic polynomial of the Magnus representation r(ϕ). In
fact, an easy calculation shows that
∂r
A= i =tI−tr(ϕ).
∂x
(cid:18) j(cid:19)
Then if we take the Fuglede-Kadisondeterminant in M(2g,Cπ), we have
detCπ tI−tr(ϕ) =detCπ tI−tr(ϕ) ∗
(cid:16) (cid:17)=detCπ(cid:16)t−1I−r(ϕ(cid:17))
because trCπ(BB∗)= trCπ(B∗B) holds. Therefo(cid:0)re the L2-tor(cid:1)sion is interpreted as
the characteristic polynomial of r(ϕ).
4. Definition of L2-torsion invariants
As was seen in Section 3, Lu¨ck’s formula gives a way to calculate the simplicial
volumefromapresentationofthefundamentalgroup. However,ingeneral,itseems
to be difficult to evaluate the exact values from the formula. In this section, we
introduce a sequence of L2-torsion invariants which approximates the original one
for a surface bundle over the circle. See [12] for details.
In order to construct such a sequence of L2-torsion invariants, we consider the
lower central series of Γ. Namely, we take the descending infinite sequence
Γ =Γ⊃Γ ⊃···⊃Γ ⊃··· ,
1 2 k
where Γ =[Γ ,Γ ] for k ≥2. Let N be the kth nilpotent quotient N =Γ/Γ
k k 1 1 k k k
−
and p :Γ→N be the natural projection.
k k
Intheprevioussection,weconsideredachaincomplexC (M ,C)ofCπ-modules.
ϕ
∗
Instead of this complex, we can use the chain complex
f
C (Mϕ,l2(π))=l2(π)⊗CπC (Mϕ,C)
∗ ∗
to define the same L2-torsion τ(M ). This point of view allows us to introduce a
ϕ
f
sequence of the L2-torsion invariants.
The groupΓ is anormalsubgroupofπ, sothatwe cantakethe quotientgroup
k
π(k)=π/Γ . Itshouldbenotedthatπ(k)isisomorphictothesemi-directproduct
k
L2-TORSION INVARIANTS AND THE MAGNUS REPRESENTATION 7
N ⋊Z. Wedenotetheinducedprojectionπ →π(k)bythesameletterp . Thereby
k k
we can consider the chain complex
C Mϕ,l2(π(k)) =l2(π(k))⊗CπC (Mϕ,C)
∗ ∗
through the projection(cid:0) pk. By using(cid:1)the Laplace operator
f
∆(k) :C M ,l2(π(k)) →C M ,l2(π(k))
i i ϕ i ϕ
on this complex, we can fo(cid:0)rmally define(cid:1)the kth(cid:0) L2-torsion in(cid:1)variant τk(Mϕ) as
follows.
Definition 4.1.
3
τk(Mϕ)= detCπ(k)(∆(ik))(−1)i+1i.
i=0
Y
Ofcourse,thisdefinitioniswell-definedifeveryL2-Bettinumberb(∆(k))vanishes
i
and every π(k) is of det≥1-class. The next lemma is easily proved(see [12], [17]).
Lemma 4.2. The L2-Betti numbers of ∆(k) are all zero.
i
Recallthe classG ofgroups. Itisthe smallestclassofgroupswhichcontainsthe
trivial group and is closed under the following processes: (i) amenable quotients,
(ii) colimits, (iii) inverse limits, (iv) subgroups and (v) quotients with finite kernel
(see [19], [24]). It is known that G contains all amenable groups. By definition,
N =Γ/Γ is a nilpotent groupand in particularan amenable group. Hence every
k k
N belongs to G. Further for any automorphism ϕ :N →N , its mapping torus
k k k
extension (HNN-extension) N ⋊Z also belongs to∗G. Therefore we have
k
Lemma 4.3. The group π(k) belongs to G.
As a result,we canconclude thatour L2-torsioninvariantsτ canbe definedfor
k
any k ≥1 and they are all homotopy invariants (see [19], [24]).
Now let us describe a formula of the kth L2-torsion invariant τ (M ) and es-
k ϕ
tablish a relation to the Magnus representation of the mapping class group. Let
p :Cπ →Cπ(k) be an induced homomorphism over the group rings. For k ≥ 1,
k
we∗put
∂r
A = p i ∈M(2g,Cπ(k)).
k k
(cid:18) ∗(cid:18)∂xj(cid:19)(cid:19)
Moreoverwe fix a constant K satisfying K ≥||R || . Then we have
k k Ak
∞
logτk(Mϕ)=−2logdetCπ(k)(RAk)
=−4glogKk+ ∞ p1trCπ(k) I −Kk−2AkAk∗ p ,
Xp=1 (cid:16)(cid:0) (cid:1) (cid:17)
by virtue of the same argument as Theorem 3.4.
For the kth invariant τ , we have taken the lower central series {Γ } of Γ and
k k
thenilpotentquotients{N }. Thesequotientsinduceasequenceofrepresentations
k
(more precisely, crossed homomorphisms)
r :M →GL(2g,ZN )
k g,1 k
for k ≥1 (see [23]). They naively approximate the original Magnus representation
r : M → GL(2g,ZΓ). By the similar observation as before, the kth invariant
g,1
8 T.KITANOANDT.MORIFUJI
logτ (M ) can be regarded as the characteristic polynomial of r (ϕ) with respect
k ϕ k
to the Fuglede-Kadison determinant in M(2g,Cπ(k)).
Fromthe viewpointofthe Magnusrepresentationofthe mappingclass group,it
seems natural to raise the following problem.
Problem 4.4. Show that thesequence{τ (M )} converges toτ(M )when we take
k ϕ ϕ
the limit on k.
In general, such an approximation problem for the L2-torsion seems to be dif-
ficult. However, similar convergence results are known for the L2-Betti numbers.
In fact, Lu¨ck shows in [18] a theorem relating L2-Betti numbers to ordinary Betti
numbers of finite coverings. This result is generalized to more general settings by
Schick in [24].
As for the Fuglede-Kadison determinant, Lu¨ck proves in [19] the following. Let
f : Q[Z] → Q[Z] be the Q[Z]-map given by multiplication with p(t) ∈ Q[Z] and
f :l2(Z)→l2(Z) be the linear operator obtainedfrom f by tensoringwith l2(Z)
(2)
over Q[Z]. Further let f : C[Z/n] → C[Z/n] be the linear operator obtained
[n]
from f by taking the tensor product with C[Z/n] over Q[Z]. We then get an
approximationresult:
logdetC[Z/n] f[n]
logdetC[Z] f(2) =nlim n
→∞ (cid:0) (cid:1)
(see [11] for a similar statemen(cid:0)t). (cid:1)In [19] Lu¨ck also points out that there exists a
purely algebraic example where Fuglede-Kadison determinants do not converge.
On the other hand, in general, we have at least an inequality for the Fuglede-
Kadison determinant in the limit statement (see [24]). That is, for the operator
R we see that
Ak
logdetCπ(RA)≥limsuplogdetCπ(k)(RAk)
k
holds. Inthelastsection,weshalldiscussProblem4.4againandgiveanaffirmative
answer under certain conditions.
5. Formulas of τ and τ
1 2
In this section, we give explicit formulasof the firsttwo invariantsof a sequence
of our L2-torsion invariants. They are really computable formulas, so that we can
makea systematic calculationfor low genus cases. Inparticular,we comparethem
withhyperbolicvolumes. Theresultsdiscussedhereareasummaryofourprevious
paper [12] (see also [10], [11]).
First we consider the Magnus representation
r :M →GL(2g,ZN ).
1 g,1 1
Here N = Γ/Γ is the trivial group and then the above representation is the
1 1
same as the usualhomologicalactionof M onH (Σ ,Z). Namely we havethe
g,1 1 g,1
representation
r :M →Aut(H (Σ ,Z),h , i)∼=Sp(2g,Z),
1 g,1 1 g,1
where h , i denotes the intersection form on the first homology group. Further
π(1)=π/Γ ∼=Z=htianditsgroupringChtiisacommutativeLaurentpolynomial
1
ring C[t,t 1]. Then the matrix A is nothing but the usual characteristic matrix
− 1
L2-TORSION INVARIANTS AND THE MAGNUS REPRESENTATION 9
of tr (ϕ). In this case, it is described by the usual determinant for a matrix with
1
commutative entries.
In order to state the theorem, we recalla definition from number theory (see [6]
and its references). For a Laurent polynomial F(t) ∈ C[t±11,...,t±n1], the Mahler
measure of F is defined by
1 1
m(F)= ··· log F(e2π√−1θ1,...,e2π√−1θn) dθ1···dθn,
Z0 Z0 (cid:12) (cid:12)
whereweassumethatundefinedt(cid:12)ermsareomitted. Namelyw(cid:12)edefinetheintegrand
(cid:12) (cid:12)
to be zero whenever we hit a zero of F.
Theorem 5.1 ([12]). The logarithm of the first invariant τ is given by
1
logτ (M )=−2m ∆ ,
1 ϕ r1(ϕ)
where ∆r1(ϕ)(t)=detA1 =det(tI −r1(ϕ)). M(cid:0)oreover(cid:1)if ∆r1(ϕ)(t) has a factoriza-
tion ∆ (t)= 2g (t−α ) (α ∈C), then we have
r1(ϕ) i=1 i i
2g
Q
logτ (M )=−2 logmax{1,|α |}.
1 ϕ i
i=1
X
Remark 5.2. Inother words,logτ (M ) is givenby the integralofthe Alexander
1 ϕ
polynomial of M over the circle S1 (see [16], for the exterior of a knot K in the
ϕ
3-sphere S3). Further, logτ (M ) can be described by the asymptotic behavior of
1 ϕ
the order of the first homology group of a cyclic covering (see [11]).
The point of the proof is to identify the Hilbert space l2(Z) with L2(R/Z) in
terms of the Fourier transforms. Then the Chti-trace trC t : l2(Z) → C can be
h i
realized as the integration
1
L2(R/Z)∋f(θ)7→ f(θ)dθ ∈C
Z0
(see [12] for details). From this description and Kronecker’s theorem ([6] Theorem
2), we obtain a certain vanishing theorem of the first invariant.
Corollary 5.3. The logarithm of τ (M ) vanishes if and only if every eigenvalue
1 ϕ
of r (ϕ)∈Sp(2g,Z) is a root of unity.
1
This corollary seems to be interesting from the viewpoint of Problem 4.4. Be-
cause in some case, we can say that the first invariant τ already approximates
1
the simplicial volume. In particular, Corollary 5.3 implies that a torus bundle
M (g = 1) with a hyperbolic structure (namely, |tr(r (ϕ))| ≥ 3) has always
ϕ 1
non-trivial L2-torsion invariant τ (M ). Summing up, we have
1 ϕ
Corollary 5.4. For any ϕ ∈ M , its mapping torus M admits a hyperbolic
1,1 ϕ
structure if and only if M has a non-trivial L2-torsion invariant τ (M ).
ϕ 1 ϕ
Therefore, the first invariant τ already approximates the simplicial volume in
1
genus one case.
Remark5.5. Itisknownthatifthecharacteristicpolynomialofr (ϕ)∈Sp(2g,Z)
1
is irreducible over Z, has no roots of unity as eigenvalues and is not a polynomial
in tn for any n>1, then ϕ is pseudo-Anosov (see Casson-Bleiler [5]). In this case,
vol(M )6=0 and further logτ (M )6=0 by Corollary 5.3.
ϕ 1 ϕ
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20 • −3πlogτ1 •
vol ⋄
5
⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄⋄-⋄
⋄•⋄•⋄•
0 50 100 |q|
Figure 1. −3πlogτ (M ) and vol(M ) vs. |q|
1 ϕ ϕ
Example5.6. Itiswell-knownthatthemappingclassgroupofthetwodimensional
q 1
torusT2 =R2/Z2isisomorphictoSL(2,Z). Takingamatrix ∈SL(2,Z),
−1 0
(cid:18) (cid:19)
it gives a diffeomorphism ϕ on T2. We may assume that it is the identity on some
embeddeddiskbyanisotopicdeformationanditgivesanelementofM . We use
1,1
the same symbol ϕ for this mapping class. An easy calculation shows that
q 1
r (ϕ)=
1 −1 0
(cid:18) (cid:19)
and
∆ (t)=det(tI −r (ϕ))=t2−qt+1.
r1(ϕ) 1
We put ξ = (q ± q2−4)/2 (the eigenvalues of the matrix r (ϕ)). If |q| ≤ 2,
1
±
then |ξ |=1. Hence logτ (M )=0 in these cases. On the other hand, either |ξ |
or |ξ |±is greater thpan one1 whϕen |q| ≥ 3, so that M has a non-trivial L2-torsio+n
ϕ
−
invariant τ in these cases. In fact, the logarithm of the first invariant is given by
1
logτ (M )=−2logmax{|ξ |,|ξ |}.
1 ϕ +
−
The values oflogτ forthe tracesq and−q arethe same,sothat itis a function
1
of |tr(r (ϕ))|. We put a graph of the L2-torsion invariant −3πlogτ (M ) and the
1 1 ϕ
hyperbolic volume vol(M ) as a function of |q| in Fig 1.
ϕ
Example5.7. Nextweconsiderthegenustwocase. Lett ,...,t betheLickorish-
1 5
Humphries generators of M2,1. We take the element ϕ = t1t3t52t−21t−41 ∈ M2,1.
As was shown in [5], the characteristic polynomial of r(ϕ) is
∆ (t)=det(tI −r (ϕ))
r1(ϕ) 1
=t4−9t3+21t2−9t+1
and irreducible over Z. Moreover it has no roots of unity as zeros. Hence, ϕ is
pseudo-Anosov and M has a non-trivial L2-torsion invariant τ (M ). In fact, we
ϕ 1 ϕ
