Sept. 18, 2008. Braids in Paris
Magnus representations of the mapping
2
class group and L -torsion invariants
Teruaki KITANO (Soka University)
Joint work with
• M. TAKASAWA-T. Morifuji, 2003–2004.
• T. Morifuji, 2006– in progress.
The main subjects of my talk;
• L2 -torsion,
– Fuglede-Kadison determinant,
• Magnus representation.
1 Determinant in Linear Algebra
Recall one of the definitions of the determinant.
Not standard, but well known in the are of zeta
function theory, dynamical systems, or spectral
geometry.
Fundamental formula:
log |det(B)|” = ”tr(log(B)).
We want to explain more precisely the above.
Most simple case: A diagonal matrix.

λ1
B=0
0
0
0
...
0 ...
...
0

0
0 .
λn
Here we assume that the eigenvalues are
0 < λ1 , . . . , λn < 1.
Directly we compute,
log(det(B)) = log(λ1 · · · λn )
=
n
X
log(λi )
=
i=1
n
X
log(1 + (λi − 1)).
i=1
Here recall the expansion of log(1 + x) at x = 0
∞
X
(−1)p+1 p
log(1 + x) =
x .
p
p=1
Then
∞
n
X
X
(−1)p+1
p
log(det(B)) =
(λi − 1)
p
p=1
i=1
!
n
∞
X X1
p
(1 − λi )
=−
p
p=1
i=1
!
n
∞
X1 X
(1 − λi )p
=−
p
p=1
i=1
∞
X
1
tr ((E − B)p ) .
=−
p
p=1
!
Hence, we can get the following equality:
log det(B) = −
∞
∑
1
p
p=1
p
tr ((E − B) ) ,
or equivalently,
(
det(B) = exp −
∞
∑
1
p
p=1
)
p
tr ((E − B) ) .
General case：
• Non diagonal matrix case：
– If B is a Hermite matrix with eigenvalues
λ1 , . . . , λn s.t. 0 < λi < 1, then the same
formula holds．
– If B is not a Hermite matrix, then we
∗
∗
replace B to BB (B :the adjoint matrix of
B)．
∗ For BB ∗ , each eigenvalue is changed
from λi of B to λi λ̄i = |λi |2 of BB ∗ .
In this case,
(
|det(B)| = exp −
∞
∑
1
1
2 p=1 p
)
∗ p
tr(E − BB )
– If some engenvalue |λi | > 1：
The problem is that the convergence radius
of log(1 + x) equals 1.
.
∗ For a sufficiently large constant K > 0
s.t. 0 < λ/K < 1, then
(
)
λ
log(λ) = log K
K
( )
λ
= log(K) + log
.
K
1
∗
∗ replace B to B (equivalently, BB to
K
1
∗
BB
)
．
K2
Summary: For any matrix B ∈ GL(n; C),
` 2 ´n
1
|det(B)| = K
exp −
2
∞
X
p=1
1
1
tr(E − 2 BB ∗ )p
p
K
This formula is one starting point to define the
Fuglede-Kadison determinant.
!
.
2 Fuglede-Kadison determinant
We extend this equality to the one in the non
commutative group algebra as the definition of
|det(B)|.
Origin: Theory of the von Neumann algebra.
• Fuglede-Kadison:Determinant theory in finite
factor, Ann. of Math. (2), 55 (1952).
In this talk, we treat only group (von Neumann)
algebra cases．
Here we fix some notations:
•
•
•
•
π: a group.
e: the unit of π.
Zπ: the group algebra of π over Z.
Cπ: the group algebra of π over C(a linear
space over C).
2
2
• l (π): l -completion
of
Cπ
，
namely
，
algebra
∑
∑
2
of all infinite sums
λg g s.t.
|λg | < ∞.
g∈π
g∈π
By using the equality log |det| = tr log, if we
can define tr, we can do det．
First the trace over Cπ is defined as follows．
Definition 2.1 Cπ-trace:
(
)
∑
trCπ
λg g = λe ∈ C.
g∈π
This Cπ-trace trCπ : Cπ → C can be naturally
extended to the trace on the matrices over Cπ.
For a matrix B = (bij ) ∈ M (n; Cπ),
trCπ (B) =
n
∑
trCπ (bii ).
i=1
By using this trace
trCπ : M (n; Cπ) → C,
Fuglede-Kadison determinant is defined as
follows.
Definition 2.2 Fuglede-Kadison determinant:
!
„
«
∞
∗ p
X
1
1
BB
2n
detCπ (B) = K exp −
trCπ E −
2 p=1 p
K2
∈ R>0 .
Here
• K > 0: a sufficiently large constant.
∗
• B = (bji ): the adjoint matrix of B = (bij ).
The adjoint matrix B ∗ is defined by the
composite of two operations;
• the complex conjugate of coefficients，
• antihomomorphism :
∑
λg g :=
∑
λg g
−1
.
Remark 2.3 The convergence of the infinite
series is not trivial! However, the following fact
is known.
If the above series for B converges, then its
L2 -betti number of B:
)
(
(
)
(
)
1
−2
∗ p
trCπ E − K BB
= 0.
lim
p→∞ p
In many cases of groups, for example, a free
group of a finite rank, an amenable group, or a
hyperbolic group, if L2 -betti number is zero,
then it converges.
3 Magnus representation
• Σg,1 : an oriented compact surface of a genus
g ≥ 1 with 1 boundary component.
• ∗ ∈ ∂Σg,1 : a base point of Σg,1 .
• Mg,1 = π0 (Diff+ (Σg,1 , ∂Σg,1 )) : the mapping
class group of Σg,1 .
• Γ = π1 (Σg,1 , ∗): free group of rank 2g.
• hx1 , . . . , x2g i: a generating system of Γ.
Proposition 3.1 (Dehn-Nielsen-Zieschang)
Mg,1 3 ϕ 7→ ϕ∗ ∈ Aut(Γ)
is injective.
The Magnus representation of the mapping class
group is defined as follows.
Definition 3.2 Magnus representation:
)
(
∂ϕ∗ (xj )
∈ GL(2g; ZΓ).
r : Mg,1 3 ϕ 7→
∂xi
i,j
Here
• ∂/∂x1 , . . . , ∂/∂x2g : ZΓ → ZΓ are the Fox’s
free differentials.
• The conjugation ∑
on ZΓ is defined as follows.
For any element
λg g ∈ ZΓ,
g
∑
g
λg g =
∑
g
λg g −1 .
Remark 3.3 This map is not a homomorphism,
but a crossed homomorphism. According to the
practice, it is called the Magnus representation
of Mg,1 .
By applying the Fuglede-Kadison determinant,
we define the characteristic polynomial for the
image of the Magnus representation.
2
4 L -torsion invariants
The problem is; How can we consider the
variable t in the characteristic polynomial ?
The answer is to consider mapping torus and the
variable ”t” of the characteristic polynomial as
the S 1 -direction element in its fundamental
group.
For the mapping class ϕ ∈ Mg,1 , we take its
mapping torus
Wϕ := Σg,1 × [0, 1]/(x, 1) ∼ (ϕ(x), 0).
From here, we put
π = π1 (Wϕ , ∗).
We fix a base point
∗ ∈ ∂Σg,1 × {0} ⊂ Σg,1 × {0} ⊂ Wϕ .
Now the group π has the following presentation:
π = hx1 , · · · , x2g , t | r1 , . . . , rn i ,
−1
−1
where ri := txi t (ϕ∗ (xi )) (i = 1, ..., 2g) and
t is the generator of π1 S 1 ∼
= Z.
Here we can consider the characteristic
polynomial
detCπ (tE − r(ϕ)) ∈ R>0
in the sense of Fuglede-Kadison determinant for
any image r(ϕ) ∈ M (2g; ZΓ) of the Magnus
representation.
By the theorem of Lück, it can be seen that it is
the L2 -torsion of the 3-manifold Wϕ for the
regular representation of π.
• L2 -torsion [Lott, Lück, Carey, Mathai, ....] is
a generalization of Reidemeister-Ray-Singer
torsion to the torsion invariant with an infinite
dimensional unitary representation.
Let us denote ρ(ϕ) by the L2 -torsion of Wϕ .
More precisely, we explain the Lück’s formula．
Applying the Fox free differentials to the relators
r1 , · · · , r2g of π, we obtain Fox matrix
(
)
∂ri
A(ϕ) :=
∈ M (2g; Zπ).
∂xj i,j
By the theorem of Lück,
log ρ(ϕ) = −2 log detCπ (A(ϕ))
= −2 log detCπ (tE − t r(ϕ))
= −2 log detCπ (tE − r(ϕ)).
Then we can say that the characteristic
polynomial of r(ϕ) is the L2 -torsion of Wϕ !
Here we want to ask the geometric meaning of
L2 -torsion：
Answer:it is the hyperbolic volume!!
Theorem 4.1 (Lott, Schick,...) For any
hyperbolic 3-manifold M ,
1
log ρ(M ) = − vol(M ).
3π
Remark 4.2 Theoretically Lück’s formula gives
the way to compute the volume of the mapping
torus Wϕ for a given ϕ.
2
5 Series of L -torsion invariants
2
It is not easy to comput L -torsion. We want to
find more computable invariant.
Fundamental framework:
Lower central series and nilpotent
quotients!
Lower central series of Γ:
Γ1 ⊃ Γ2 ⊃ · · · ⊃ Γk ⊃ · · ·
•
•
•
•
•
Γ1 = Γ,
Γ2 := [Γ1 , Γ1 ],
Γk := [Γk−1 , Γ1 ] (k ≥ 2).
Nk := Γ/Γk :k-th nilpotent quotient
pk : Γ → Nk : natural projection.
Under these situations, we obtain the series of
(graded) Magnus representations:
rk : Mg,1 → GL(2g; ZNk ).
In the case of k = 1, the projection is
p1 : Γ = π1 (Σg,1 ) → N1 = {e},
then we obtain the map
r1 : Mg,1 → GL(2g; Z),
which is just the homology representation of the
mapping class group.
In the case of k = 2, the projection is
Γ = π1 (Σg,1 ) → H = H1 (Σg,1 ; Z),
then we obtain the map
r2 : Mg,1 → GL(2g; ZH).
If we restrict this map to the Torelli group
Ig,1 = Ker{Mg,1 → Sp(2g; Z)},
then
r2 : Ig,1 → GL(2g; ZH)
is a homomorphism.
Because Γk C Γ, then
π(k) = π/Γk ∼
= Nk o Z.
Then
pk : π → π(k) = Nk o Z,
and we write pk ∗ : Cπ → Cπ(k) to its induced
homomorphism on the group ring. The k-th Fox
matrix
(
(
))
∂ri
Ak := pk ∗
∈ M (2g; Cπ(k)).
∂xj
[Morifuji-Takasawa-Kitano].
For Wϕ , k-th L2 -torsion invariant ρk (ϕ) can be
defined and ，
log ρk (ϕ) = −2 log detCπ(k) (Ak ).
Remark 5.1 The k-th invariant ρk (ϕ) can be
counted the characteristic polynomial of a
matrix rk (ϕ) ∈ GL(2g; Cπ(k)) in terms of
Fuglede-Kadison determinant.
The first and fundamental problem is the
following.
Problem 5.2
lim (ρk (ϕ)) = ρ(ϕ)?
k→∞
It is still the open problem. We explain some
computation and some partial results about
these invariants.
6 Example
We compute the first and second invariant for
the torefoile knot exterior. Before doing this, we
can see the following by the general theory of
torsion invariants.
Proposition 6.1 log ρk (ϕn ) = n log ρk (ϕ).
Remark 6.2 Topologically, Wϕn → Wϕ is an
n-fold cyclic covering of Wϕ .
Because L2 -torsion is a topological invariant of
mapping torus, then we can see the following.
Proposition 6.3 For any mapping class
ϕ ∈ Ker{Mg,1 → Mg }, it holds that
log ρk (ϕ) = 0.
Remark 6.4 For ϕ as above, topologically
1
∼
Wϕ = Σg,1 × S .
Now we consider the torefoile knot exterior
E(31 ).
First E(31 ) admits a strcture of Σ1,1 -bundle
over S 1 . Then there exists ϕ ∈ M1,1 s.t.
E(31 ) = Wϕ .
We can choice ϕ s.t. ϕ6 is the Dehn twist along
the boudary parallel simple closed curve, that is,
ϕ6 ∈ Ker(M1,1 → M1 ).
On π1 (Σ1,1 ) = hx1 , x2 i, we can do that
ϕ6∗ (x1 ) = [x1 , x2 ]x1 [x1 , x2 ]−1 ,
ϕ6∗ (x2 ) = [x1 , x2 ]x2 [x1 , x2 ]−1 .
Then
π1 (Wϕ6 ) = hx1 , x2 , t| tx1 t−1 = ωx1 ω −1 ,
tx2 t−2 = ωx2 ω −1 i
where ω = [x1 , x2 ] =
−1 −1
x1 x2 x1 x2 .
For this case, we can directly compute the Fox
free differentials
∂
−1
−1
(tx1 t − ωx1 ω ),
∂x1
∂
(tx2 t−1 − ωx2 ω −1 ),
∂x2
and obtain the Fox matrix A(ϕ6 ) as follows.
(
a11
6
A(ϕ ) =
a21
a12
a22
)
∈ GL(2; Zπ1 (Wϕ6 ))
where
−1
a11 = t − 1 + xyx
− ω + ωxω
a12 = −x + ω + ωxω
−1
a21 = −1 + xyx
a22 = t − x + ωyω
−1
−1
(1 − xyx
(x − ω)
+ ωyω
−1
−1
−1
−1
(1 − xyx
(x − ω).
)
)
• Computation of ρ:
It is hard to compute the Fuglede-Kadison
determinant by definition. However we know
that Wϕ = E(31 ) and Wϕ6 do not admit
hyperbolic structures. Then
log ρ(ϕ) = log ρ(ϕ6 ) = 0.
• k = 1 case:
In this case, by applying the projection
π = π1 (Wϕ6 ) → π(1) = hti, we can obtain
(
)
t−1
0
6
A1 (ϕ ) =
.
0
t−1
Then
log ρ1 (ϕ6 ) = −2 log detCπ(1) (A1 (ϕ6 ))
= −2 log detCπ(1) (tE − E)
= 0 by computation.
On the other hand, we have seen that
6
log ρ1 (ϕ ) = 6 log ρ1 (ϕ) = 0.
Therefore we can see
log ρ1 (ϕ) = log ρ1 (E(31 )) = 0.
• k = 2 case:
In this case, we can reduce A(ϕ6 ) to
)
(
2
t + x + y − xy − 2
x − 2x + 1
6
.
A2 (ϕ ) =
2
−y + 2y − 1
t + xy − x − y
Then
log ρ1 (ϕ6 ) = −2 log detCπ(2) (A2 )
= 0 by computation.
On the other hand,
log ρ2 (ϕ6 ) = 6 log ρ2 (ϕ) = 0.
Then
log ρ2 (ϕ) = log ρ2 (E(31 )) = 0.
In the genus 1 case, the following holds for any
mapping class.
Theorem 6.5 (MTK)
log ρ1 (ϕ) = −2 log max{1, |α|, |α|
−1
}.
Here
α, α−1 : the eigenvalues of ϕ∗ ∈ SL(2; Z).
Here we consider the following family of
matricies
)
(
a 1
ϕ∗ =
−1 0
where tr(ϕ∗ ) = a.
We compute ρ1 and values of hyperbolic volume
(by using SnapPea) and compare them.
tr(r1 (ϕ)) = a
0
1
2
3
4
5
6
7
8
−3π log ρ1 (ϕ)
0
0
0
18.1412
24.8240
29.5334
33.2270
36.2825
38.8948
volume
0
0
0
2.0298
2.6667
2.9891
3.1772
3.2969
3.3775
In the case of k = 2, we can prove the following.
Theorem 6.6 (MTK) For any ϕ ∈ M1,1 ,
log ρ2 (ϕ) = 0.
Furthermore we can prove the following.
Theorem 6.7 (MK) If ϕ ∈ M1,1 is a Dehn
twist which is a twist along a non-separating
curve(=not parallel to the boundary), then
log ρk (ϕ) = 0.
We remark that vol(Wϕ ) = 0 for such ϕ.
Therefore it is trivial in some sense, but it holds
that
log ρk (ϕ) = log ρ(ϕ)
for such ϕ.
Remark 6.8 We have not yet computed the
third invariant ρ3 for non trivial examples. It is
very hard to compute!
7 Results
Recall Nielsen-Thurston classification of the
mapping classes:
• periodic
• reducible
• pseudo Anosov
In Mg,1 , there is no periodic element. However，
for a lift of a periodic element in Mg ,
i.e., for any mapping class ϕ ∈ Mg,1 such that
ϕn ∈ Ker{Mg,1 → Mg }, we can see the
following.
Proposition 7.1 For any ϕ s.t. ϕn ∈ Ker, then
log ρk (ϕ) = 0 .
Proposition 7.2 For ϕ ∈ Mg,1 , if there exists a
separating simple closed curve γ ⊂ Σg,1 such
that ϕ fixes pointwisely γ ，then ρk (ϕ) can be
computed by the both of L2 -torsion invariants
of 2 compact surfaces cutted by γ.
Remark 7.3 In the above case, we can reduce
computation to the one for lower genus cases.
k = 1 case,
• π(1) = π/Γ1 = N1 o Z ∼
=Z
• GL(2g; Zπ(1)) = GL(2g; Z).
Here Magnus representation is just
r1 : Mg,1 → Sp(2g : Z).
This Fuglede-Kadison determinant is the usual
determinant over C[Z]. ρ1 can be computed as
follows.
Theorem 7.4 (Lott, Lück, MTK) For any
ϕ ∈ Mg,1 ,
∫
log ρ1 (ϕ) =
log | det(tE − r1 (ϕ))|dt.
S1
Remark 7.5 This integration is the Mahler
measure for 1-variable polynomials.
By properties of Mahler measure, we can see the
following.
Corollary 7.6
log ρ1 (ϕ) = −2
2g
∑
log max{1, |αi |}
i=1
Here
α1 , . . . , α2g : the eigenvalues of r1 (ϕ) ∈ Sp(2g; Z).
In the genus 1 case, 2 eigenvalues of
r1 (ϕ) ∈ SL(2; Z) can be determined by the
trace. Now we see the following corollary.
Corollary 7.7 It holds;
log ρ1 (ϕ) = 0 ⇔ |tr(r1 (ϕ))| < 2.
Recall that −3π log ρ(ϕ) = vol(Wϕ ), then now
we compare the volume with −3π log ρ1 (ϕ).
Problem 7.8 For A such that |tr(A)| > 2,
compute ρk and investigate its behavior when
k → ∞．
Higher genus case：
k = 2, Magnus representation
r2 : Ig,1 → GL(2g; ZH).
The mapping class ϕ∗ acts on H trivially,
π(2) = H × Z,
then Cπ(2) is a commutative ring. Hence, we
can use the usual determinant.
Under this situation,
Theorem 7.9 (MTK) log ρ2 can be described
by using the Mahler measure for multi-variable
polynomials.
From computation by M. Suzuki,
Corollary 7.10 If ϕ is a BP-map, or a
BSCC-map, then
log ρ2 (ϕ) = 0.
Remark 7.11 There exists ϕ ∈ Ig,1 such that
log ρ2 (ϕ) 6= 0.
Theorem 7.12 (MK) If ϕ ∈ Mg,1 is a product
of Dehn twists along any disjoint non-separating
simple closed curves which are mutually
non-homologous, then
log ρk (ϕ) = 0.