Complex lambda length as parameter for SL(2, C)
representation spaces of punctured surface groups
Toshihiro Nakanishi, Shimane Univeristy
Introduction
Let Fg,n be an oriented closed surface of genus g
with n punctures at x1 ,..., xn .
The fundamental group π1 (Fg,n ) has a canonical
generator system
(a1 , b1 , ..., ag , bg , d1 , ..., dn )
satisfying
[a1 , b1 ] · · · [ag , bg ]d1 · · · dn = 1,
where dj represents a loop which goes around
the jth puncture xj in the positive direction.
Introduction
In this talk, it is always assumed that 2g − 2 + n > 0.
SL(2, C) =
��
a b
c d
�
�
: a, b, c, d ∈ C, ad − bc = 1
An element P of SL(2, C) is parabolic if P is conjugate to
�
�
1 1
±
0 1
P = {P ∈ SL(2, C) : P is parabolic and trP = −2}
�
�
−1 −1
= the conjugacy class of
in SL(2, C).
0 −1
Introduction
Let
Rg,n =
�
ρ is faithful representation
ρ : π1 (Fg,n ) → SL(2, C) :
and ρ(dj ) ∈ P, j = 1, ..., n
Remark. If ρ : π1 (Fg,n ) → P SL(2, C) is a faithful and
discrete representation such that ρ(dj ) is parabolic for
j = 1, ..., n, then there is a lift ρ̃ : π1 (Fg,n ) → SL(2, C)
of ρ which belongs to Rg,n (Kra 1985)
Purpose of this talk is:
to introduce a coordinate-system to Rg,n .
�
/ ∼
conj
Penner’s λ-length coordinates
For the punctured surfaces Fg,n , n > 0, we employ
R.C.Penner’s lambda lengths
Penner [1] introduced a coordinate-system by λ-lengths of
ideal arcs on Fg,n to the decorated Teichmüller space
T̃ (g, n), and obtained a cell decomposition of T̃ (g, n)
invariant under the action of the mapping class group
M Cg,n .
[1] Penner, R. C., The decorated Teichmüller space of punctured surfaces,
Comm. Math. Phys. 113 (1987), no. 2, 299–339.
Penner’s λ-length coordinates
For each ideal triangulation ∆ = (c1 , c2 , ..., cD ) of Fg,n ,
D = 6g − 6 + 3n, the tuple of their λ-lengths
(λ(c1 ), λ(c2 ), ..., λ(cN ))
gives a homeomorphism λ∆ : T̃ (g, n) → RD
+.
For two ideal triangulations ∆ and ∆� ,
T̃ (g,
 n)

id�
T̃ (g, n)
λ∆
−−−−→
λ ∆�
−−−−→
the coordinate change λ∆� ◦
λ∆ (T̃(g, n)) =

−1
λ
◦λ
�
� ∆ ∆
D
R+
λ∆� (T̃ (g, n)) = RD
+
−1
λ∆
is a rational transformation.
Penner’s λ-length coordinates
In particular, if ϕ : Fg,n → Fg,n is an orientation
preserving homeomorphim, then
ϕ∗ (λ(c1 ), ..., λ(cD )) = (λ(ϕ
−1
(c1 )), ..., λ(ϕ
−1
(cD )))
is a rational transformation.
So the mapping class group acts on RD
+ as a group of
rational transformations.
Since the Teichmüller space T (g, n) is naturally embedded
in T̃ (g, n), the coordinates can also parametrize T (g, n).
Example Once Punctured Torus Group
Let A and B be a canonical generator system of
a once punctured torus group, normalized so that
AB(∞) = 0
By some examples, we see how λ-length coordinates recover
and
a generator system of a�surface group
�
=
−1
0
λ2 λ3
λ1
− λ2
3
−λ1
λ2
λ3

−1
ABA
B
−1
Then


A=
λ21 +λ23

,
λ-length
1
−1
= P −1 ,


B=
−
λ22 +λ23
λ1 λ3
−λ2
λ2
− λ2
3
λ1
− λ3


.
λ-length


AB = 

2
λ1
+ +
λ1 λ2
2
λ2
2
λ3
λ3

1
−

λ3 

0
λ-length
Here we used λ-lengths defined by
λ1 = −trA − trB AB, λ2 = −trB − trA
λ3 = trAB + trBA,
−1
satisfying
2
λ1
+
2
λ2
+
2
λ3
=
1
2 λ1 λ2 λ3 .
−1
BA
The mapping class group MC 1,1 of the once punctured
torus is identified with a subgroup of the group of outer
automorphisms of the once punctured group G.
Consider the group automorphisms ϕ1 and ϕ2 of
G determined by the following transformations on the
generator system:
ϕ1 : (A, B) �→ (A, BA),
ϕ2 : (A, B) �→ (AB, B),
These automorphisms acts on λ-length coordinates by
�
�
λ21 + λ23
ϕ1∗ (λ1 , λ2 , λ3 ) = λ1 , λ3 ,
,
λ2
ϕ2∗ (λ1 , λ2 , λ3 ) =
�
λ3 , λ 2 ,
λ22
+
λ1
λ23
�
.
However, it is more convenient to use
λ1 = − 16 (trA + trB −1 AB), λ2 = − 16 (trB + trA−1 BA)
λ3 =
1
6 (trAB
+ trBA),
which satisfy the Markoff equation
2
λ1
2
λ2
+
+
P =
�
In this case
2
λ3
= 3λ1 λ2 λ3 .
−1
0
6
−1
�
.
About McShane’s identity for once puncture torus groups
McShane’s identity for a once punctured torus with
a complete, finite volume hyperbolic structure is
�
γ
1
1
=
l(γ)
2
1+e
The sum is over all simple closed geodesics γ on the torus,
and l(γ) denotes the hyperbolic length of γ.
McShane’s identity
Let {A, B} be a canonical generator system of a once
punctured torus group G. Let
G0 = {(A, B), (B −1 , A)}
and define Gn (n = 1, 2, ...) by induction: (g, h) ∈ Gn
if and only if there exist a pair (a, b) in Gn−1 and
a positive integer k such that either
(g, h) = ((ab)
k−1
Let
a, (ab) a), or (g, h) = ((ba) b, (ba)
k
G=
k
∞
�
n=0
Gn .
k−1
b).
McShane’s identity
Then McShane’s identity can be expressed as
�
� |tr(gh)| − (trgh)2 − 4
= 1.
|tr(gh)|
(g,h)∈G
However, if the canonical generator system (A, B)
is chosen so that trA < −2 and trB < −2, then
�
� tr(gh) − (trgh)2 − 4
= 1.
tr(gh)
(g,h)∈G
McShane’s identity
Let G be the once punctured torus group in SL(2, Z)
with canonical generators




−2
1
−2 −1
, B = 
.
A=
1 −1
−1 −1
McShane’s identity in this case would be
√
� 3m − 9m2 − 4
√
√
(3 − 5) + (3 − 2 2) + 2
= 1,
m
m
where m runs over all Markoff numbers ≥ 5:
m = 5, 13, 29, 34, 89, 169, 194, · · · , if a conjecture
about Markoff numbers were settled.
For the partial sum for m from 5 to 610, the value is
0.9999898618 · · ·
Why we need complex lambda lengths?
Example: Twice punctured torus
Let (A, B, C, D), ABA−1 B −1 CD = 1, be a canonical
generator system of a twice punctured torus group G
in SL(2, C).
Define
λ1
λ2
λ3
λ4
λ5
= −trA − trBA B C,
−1
−1
= trABA + trB C,
= −trA−1 − trB −1 CAB,
= −trAB − trA−1 B −1 C,
= −trC − trABA−1 B −1 .
−1
−1
Why we need complex lambda length?
Then under the normalization D(∞) = ∞ and C(∞) = 0,


λ2 λ4 λ25 + λ3 λ24 λ5 + λ22 λ3 λ5 + λ1 λ23 λ5 + λ2 λ23 λ4
−

λ1 λ2 λ4 λ25

,

λ3
λ5

2
λ2 λ5 + λ1 λ3 λ5 + λ2 λ3 λ4
−

λ1 λ4 λ25

,

2
2
2
2
λ2 λ4 λ5 + λ1 λ2 λ5 + λ3 λ4 λ5 λ1 λ3 λ5 + λ1 λ2 λ3 λ4 
λ1 λ3 λ4 λ5
λ24 λ5 + λ22 λ5 + λ1 λ3 λ5 + λ2 λ3 λ4

λ2 λ4 λ5

A=

λ1
λ-length

λ2
−
λ5


B=

 λ1 λ2 + λ4 λ5
λ3


C=
0
λ5
1 
−
λ5 
,
λ-length
−2
λ-length
Why we need complex lambda length?
Here
if
λ3 λ5 (λ21 + λ22 + λ24 ) + λ1 λ5 (λ22 + λ23 + λ24 ) + λ2 λ4 (λ21 + λ23 + λ25 )
= (λ5 − 2)λ1 λ2 λ3 λ4 ,
D
−1
= ABA
−1
B
−1
C=
�
−1
0
1
−1
�
.
Remark. We can replace the identity above by
λ3 λ5 (λ21 + λ22 + λ24 ) + λ1 λ5 (λ22 + λ23 + λ24 ) + λ2 λ4 (λ21 + λ23 + λ25 )
= (11λ5 − 2)λ1 λ2 λ3 λ4 ,
so that this has a solution
(λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 , λ8 , λ9 ) = (1, 1, 1, 1, 1, 1, 1, 1, 1)
In this case
D
−1
= ABA
−1
B
−1
C=
�
−1 11
0 −1
�
.
Why we need complex lambda length?
The mapping class group MC 1,2 of the twice punctured
torus is identified with a subgroup of the group of outer
automorphisms of G.
Consider the group automorphisms ϕ1 , ϕ2 and ϕ3 of
G determined by the following transformations on the
generator system:
ϕ1 : (A, B, C) �→ (A, BA, C),
ϕ2 : (A, B, C) �→ (ABA−1 , A−1 , C),
ϕ3 : (A, B, C) �→ (A, BA−1 B −1 CB, BA−1 B −1 CBAB −1 ).
Why we need complex lambda length?
These automorphisms acts on λ-length coordinates by
ϕ1∗ (λ1 , λ2 , λ3 , λ4 , λ5 ) =
ϕ2∗ (λ1 , λ2 , λ3 , λ4 , λ5 ) =
�
�
λ1 , λ 4 , λ 3 ,
λ2 , λ 3 ,
λ24
λ22 λ5
�
+ λ1 λ3
, λ5 ,
λ2
+ λ1 λ3 λ5 + λ2 λ3 λ4
,
λ1 λ4
λ22
�
+ λ1 λ3
, λ5 ,
λ4
and
ϕ3∗ (λ1 , λ2 , λ3 , λ4 , λ5 )
�
λ2 λ4 λ5 + λ1 λ22 + λ21 λ3 λ1 λ24 λ5 + λ2 λ4 λ25 + λ21 λ2 λ4 + λ1 λ22 λ5 + λ21 λ3 λ5
= λ1 ,
,
,
λ3 λ4
λ2 λ3 λ4
�
2
2
2
λ1 λ2 + λ4 λ5 (λ4 λ5 + λ1 λ2 λ4 + λ1 λ2 λ4 λ5 + λ2 λ5 + λ1 λ3 λ5 + λ2 λ3 λ4 )
,
.
2
2
λ3
λ2 λ4 λ5
Why we need complex lambda length?
We consider
ψ=
−1
ϕ1 ϕ3 ϕ2
: (A, B, C) �→ (ABA
−1
,C
−1
2
−1
B A
,C
−1
BCB
−1
C)
Then the transformation ψ∗ induced by ψ fixes the point
√
√
τ0 = (−4, 4, −4 + 8 −1, −8, −8 − 8 −1).
If (A, B, C) ∈ R1,2 be the point which has the coordinates
τ0 , then there is M ∈ SL(2, C) such that
(ABA−1 , C −1 B 2 A−1 , C −1 BCB −1 C) = (M −1 AM, M −1 BM, M −1 CM ).
This fixed point can be found, because complex
lambda lengths are treated.
Why we need complex lambda length?
The group Γ = �A, B, C, M � acts on the 3-dimensional
hyperbolic space H3 and H3 /Γ is the Whitehead link
complement. This space is fibered over the circle with
fiber twice punctured torus.
from Wikipedia
Example Once Punctured Genus 2 Surface Group
Let (A, B, C, D) be a canonical generator system of
a once punctured genus 2 surface group with
−1 −1
−1 −1
P = ABA B CDC D parabolic. Let
x1 = −tr(DCD−1 ) − tr(DCD−1 P ),
x2 = tr(DCD−1 C −1 D−1 ) − tr(DCD−1 C −1 D−1 P ),
x3 = −tr(C −1 D−1 ) − tr(C −1 D−1 P ),
x4 = −tr(DC 2 DC −1 D−1 ) − tr(DC 2 DC −1 D−1 P ),
x5 = −tr(BAB −1 A−1 ) − tr(BAB −1 A−1 P ),
x6 = −tr(AB −1 A−1 ) − tr(AB −1 A−1 P ),
x7 = −tr(ABAB −1 A−1 ) − tr(ABAB −1 A−1 P ),
x8 = −tr(AB) − tr(ABP ),
x9 = −tr(ABA−1 B −2 A−1 ) − tr(ABA−1 B −2 A−1 P ).
Then


A=


B=


C=


D=
−
x25 x6 x7 +x5 x27 x8 +x5 x26 x9 +x6 x7 x8 x9 +x5 x28 x9
x5 x6 x8 x9
−
x5 x27 +x6 x7 x9 +x5 x8 x9
x25 x6 x8
x5 x6 +x7 x8
x9
x7
x5
x6
x5
x5 x7 +x6 x9
x25 x8
x5 x26 +x6 x7 x8 +x5 x8 x9
−
x7 x9
x25 x6 x7 +x5 x27 x8 +x5 x26 x9 +x6 x7 x8 x9 +x5 x8 x29
−
x5 x7 x8 x9
x1
x5
2 x5
− x1 xx43+x
x25
x1 x2 x3 +x21 x5 +x3 x4 x5
x2 x4
x1 x2 x3 x4 +x22 x3 x5 +x21 x4 x5 +x3 x24 x5 +x1 x2 x25
−
x2 x3 x4 x5
x1 x2 x3 x4 +x22 x3 x5 +x21 x4 x5 +x23 x4 x5 +x1 x2 x25
−
x1 x3 x4 x5
x1 x2 x4 +x22 x5 +x3 x4 x5
x1 x3 x25
1 x5
− x2 x3x+x
4
x2
x5












where
−1
ABA
B
−1
CDC
−1
if
D
−1
=
�
−1
0
1
−1
�
,
x1 x2 x3 x4 x5 x6 x7 x8 x9
= x1 x2 x3 x4 x25 x6 x7 + x1 x2 x3 x4 x5 x26 x8 + x1 x2 x3 x4 x5 x27 x8
+x1 x2 x3 x4 x6 x7 x28 + x1 x2 x3 x4 x5 x26 x9 + x1 x2 x3 x4 x5 x27 x9
+x1 x2 x23 x6 x7 x8 x9 + x1 x2 x24 x6 x7 x8 x9 + x21 x3 x5 x6 x7 x8 x9
+x22 x3 x5 x6 x7 x8 x9 + x21 x4 x5 x6 x7 x8 x9 + x22 x4 x5 x6 x7 x8 x9
2
+x1 x2 x5 x6 x7 x8 x9
+x1 x2 x3 x4 x5 x8 x29
+
2
x1 x2 x3 x4 x5 x8 x9
+
2
x1 x2 x3 x4 x6 x7 x9
Definition of lambda length
Definition of λ-length
If P1 and P2 in P do not commute, then there is
Q ∈ SL(2, C) such that
P1 P2 = −Q .
2
This Q necessarily satisfies
P2 = Q
−1
P1 Q.
In what follows, the diagram
Q
P1 −
→ P2
means that P1 , P2 ∈ P do not commute and Q2 = −P1 P2 .
Definition of lambda length
Define
λ(P1 , P2 ) = trQ.
Remark. λ(P1 , P2 ) is defined up to sign.
If P2 = A−1 P1 A for A ∈ SL(2, C), then
λ = λ(P1 , P2 ) = ±(trA + trAP1 )
�
�
−1
1
If AB =
= P −1 , then
0 −1
A=
�
∗
λ
∗
∗
�
, B=
−trA − trAP
�
∗
λ
∗
∗
�
−trB − trBP
Definition of lambda length
P1
In this diagram,
Q3
P2
P2 =
−1
Q3 P1 Q3 , P3
=
Q2
Q1
−1
Q1 P2 Q1 , P1
P3
=
−1
Q2 P3 Q2 .
Thus, Q2 Q1 Q3 commutes with P1 and hence
trQ2 Q1 Q3 = ±2.
If trQ2 Q1 Q3 = −2, then (Q1 , Q2 , Q3 ) is called
a (-)-system.
Definition of lambda length
Ideal Ptolemy Identity
Consider the following diagram.
We assume that
trQ5 = trQ�5 , trQ6 = trQ�6 ,
which means that
�
Q5
=
−1
P1 Q 5 P1 ,
�
Q6
=
−1
P4 Q 6 P4 .
Definition of lambda length
If three of
(Q1 , Q2 , Q5 ), (Q�5 , Q3 , Q4 ), (Q1 , Q6 , Q4 ), (Q2 , Q3 , Q�6 )
are (−)-systems, then the remaining one is also
a (−)-system.
In this case
trQ1 trQ3 + trQ2 trQ4 = trQ5 trQ6 .
Ideal Ptolemy identity
Definition of lambda length
An ideal arc on Fg,n is an arc connecting two (not
necessarily distinct) punctures.
An ideal triangulation (c1 , ...., cD ) on Fg,n is a
triangulation of Fg,n by simple ideal arcs.
By using the Euler characteristic, it is easy to see
D = 6g − 6 + 3n
An ideal triangulation of the once punctured genus 2 surface
Definition of lambda length
It is possible that a component of the ideal triangulation
(a triangle) looks like the figure below.
Let Γ be a Fuchsian group of type (g, n) and identify Fg,n
with H/Γ.
Γ = �A1 , B1 , ..., Ag , Bg , D1 , ..., Dn �
where
[A1 , B1 ] · · · [Ag , Bg ]D1 · · · Dn = 1
and
D1 , ..., Dn ∈ P
Definition of lambda length
Let c be an ideal arc and choose a lift c̃ of c in H. Then
The end points of c̃ are fixed points of parabolic elements
P1 and P2 , which are conjuate with some Dj , respectively
in Γ.
Let [ρ] ∈ Rg,n , where we identify π1 (Fg,n ) with Γ.
Then ρ(P1 ), ρ(P2 ) ∈ P, and they do not commute.
Define
λ(c, ρ) = λ(ρ(P1 ), ρ(P2 ))
Remark. λ(c, ρ) is defined up to sign.
Definition of lambda length
Let ∆ = (c1 , ..., cN ) be an ideal triangulation.
A tuple of λ-lengths
(λ(c1 , ·), ..., λ(cN , ·)) : Rg,n → CN
is subordinate to (−)-systems if
for each triangle (ci , cj , ck ) in ∆, the triple (Q3 , Q1 , Q2 )
with
λ(ci ) = trQ1 , λ(cj ) = trQ3 , λ(ck ) = trQ3 ,
(Q3 , Q2 , Q1 ) is a (−)-system.
P1
Q3
P2
Q2
Q1
P3
Definition of lambda length
For each ideal triangulation ∆, there is a choice of λ-lengths
so that
(λ(c1 , ·), ..., λ(cN , ·)) : Rg,n → CN
is subordinate to (−)-systems.
Remark. The image of
(λ(c1 , ·), ..., λ(cN , ·))
is contained in the intersection of zero sets of
n polynomial equations.
Expression of the λ-length of an ideal arc
Let ∆ = (c1 , c2 , ..., cd ), d = 6g − 6 + 3n, be an ideal
triangulation of Fg,n , and λj = λ(cj ) be
the λ-length coordinate associated to cj .
We assume that (λ1 , ..., λd ) is subordinate to (−)-systems.
If c is an ideal arc, then
where
Pc (λ1 , ..., λd )
λ(c) = a1 a2
ad ,
λ 1 λ2 · · · λ d
so that we can apply
the ideal Ptolemy
Pc (λ1 , ..., λd )
is a homogeneous polynomial with positive integer
coefficients of degree a1 + a2 + · · · + ad + 1, and
aj is the geometric intersection number of c and cj .
The expression above is not necessarily in the
lowest terms.
We consider only once punctured surfaces Fg,1 , g ≥ 1.
In this case, if c is simple, then the expression
Pc (λ1 , ..., λd )
λ(c) = a1 a2
ad ,
λ 1 λ2 · · · λ d
in the lowest terms is with a1 ,..., ad (d = 6g − 5 + 2n)
defined as follows:
The blue lines are the lift of ∆
in H, and the green line of c̃.
Remark. For the case of once punctured torus, aj is
the geometric intersection number of c and cj .
Closed surface of genus 2
We identify the Teichmüller space Tg (g ≥ 2) with the
space of marked Fuchsian groups.
A point of Tg is a simultaneous conjugacy class of
a generator system
(A1 , B1 , ...., Ag , Bg )
satisfying the relation
[A1 , B1 ] · · · [Ag , Bg ] = 1.
We fix a Fuchsian group Γ0 as a reference point. A point
of Tg is represented also by a conjugacy class of Fuchsian
representations
ρ : Γ0 → SL(2, C).
Each element g of Γ0 defines a function (the trace function)
by
χ g : Tg → R
χg (ρ) = trρ(g).
Theorem (Schmutz 1993, Okumura 1996, Feng Luo 1998,
Hamenstädt 2003)
There are d = 6g − 5 elements g1 ,..., gd of Γ0 such that
the tuple of their absolute trace functions
embeds Tg into
d
R+ .
(|χg1 |, ..., |χgd |)
Remark It is impossible to embed Tg into R6g−6 by any
tuple of 6g − 6 trace functions (Wolpert).
For g = 2, let (A, B, C, D) be a point of T2 such that
trA > 0, trB > 0, trC > 0, trD > 0.
Then the following seven traces (which are all positive)
embed T2 into R7+ .
a = trA,
v = −trACD2 ,
b = trB,
w = −trACD,
z = trAB,
t = trCD
u = −trACDC −1
.
They satisfy
awt + a2 + w2 + t2 + K 2 + S 2 + 4
�
−w (K 2 + 4)(S 2 + 4) = 0,
where
�
�
K = abz − a2 − b2 − z 2 , S = uvt − u2 − v 2 − t2
Lemma. Let G be a group generated by n elements
A1 , A2 , ...., An
in SL(2, C), then for any element X ∈ G, trX is a
polynomial with integer coefficients in the variables
�
trAj1 Aj2 · · · Ajk
1 ≤ j1 < j2 < · · · < jk ≤ n,
:
k = 1, ..., n
�
For the Fuchsian group �A, B, C, D� = �A1 , A1 , A2 , A4 �,
each element of
�
�
1 ≤ j1 < j2 < · · · < jk ≤ 4,
trAj1 Aj2 · · · Ajk :
k = 1, ..., 4
is a rational function of
a = trA,
v = −trACD2 ,
b = trB,
w = −trACD,
z = trAB,
t = trCD
u = −trACDC −1
.
(K 2 + S 2 + t2 + a2 + 4)u + w(2atu − 2av − uw + t2 uw − tvw)
trC =
,
2
2
w(S + t )
(K 2 + S 2 + t2 + a2 + 4)v + w(2au + twu − vw)
trD =
.
2
2
w(S + t )
uw(2a + tw) + v(4 + a2 + K 2 − w2 )
trAC = −
,
2
2
S +t
(4 + a2 + K 2 − w2 )v + wu(2a + tw)
,
trAD = (ad + u − cw) + t
2
2
S +t
c(2b + a2 b − 2az + bK 2 ) − tuz + dw(ab + z + zK 2 ) − v(ab + zK 2 )
trBC =
,
2
2
K +a
2(adz − bd) − u(ab + K 2 z) + tv(ab + z + K 2 z) + (c − dt)w(ab + z + K 2 z)
trBD =
,
2
2
K +a
−2cz − btu + avz + wd(b − az)
trABC =
,
2
2
K +a
d(K 2 + a2 + 2) + auz + vt(b − az) + w(bc − bdt − acz + adtz)
trABD =
,
2
2
K +a
t(2b + a2 b − 2az + bK 2 ) + dvz + w(ab + K 2 z) + u(cz − dtz)
trBCD =
,
2
2
K +a
−2tz + b(c − dt)u + bdv − awz
trABCD =
K 2 + a2
So for each X ∈ Γ = �A, B, C, D�, trX is a rational function
of a, b, z, u, v, w, t.
A mapping class ϕ induces an outer automorphism ϕ∗ of Γ.
and
(a, b, ..., w, t) = (trA, trB, ..., −trACD, trCD)
↓ ϕ∗
(trϕ∗ (A), trϕ∗ (B), ..., −trϕ∗ (ACD), ϕ∗ (CD))
is a rational transformation.
One may need to change the sign of elements.
As a corollary, we can conclude:
The mapping class group M C2 is represented by a group
Mercifin
beaucoup
of rational transformations
in
the variables. a, b, z, u, v,
w and t.
The speaker thanks the organizers of the Nara conference, Teruaki Kitano, Takayuki Morifuji,
Ken’ichi Ohshika and Yasushi Yamashita.
He also thanks Marjatta Näätänen and Gou Nakamura.
fin