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∗Teichmu¨ller theory

Louis Funar Rinat M. Kashaev

Institut Fourier BP 74, UMR 5582 Section de Math´ematiques, Case Postale 64

University of Grenoble I Universit´e de Gen`eve

38402 Saint-Martin-d’H`eres cedex, France 2-4, rue du Li`evre, 1211 Geneve 4, Suisse

e-mail: funar@fourier.ujf-grenoble.fr e-mail: rinat.kashaev@unige.ch

March 22, 2010

Abstract

The central extension of the mapping class groups of punctured surfaces of ﬁnite type that arises in

quantum Teichmu¨ller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is

analogous to the result obtained in [9] for the Thompson groups.

2000 MSC Classiﬁcation: 57M07, 20F36, 20F38, 57N05.

Keywords: Mapping class group, Ptolemy groupoid, quantization, Teichmu¨ller space, Meyer class, Euler

class.

Introduction

ThequantumtheoryofTeichmu¨llerspacesofpuncturedsurfacesofﬁnitetype,originallyconstructedin[4,15]

and subsequently generalizedto higher rank Lie groups and cluster algebras in [7, 8], leads to one parameter

familiesofprojectiveunitaryrepresentationsofPtolemymodulargroupoidsassociatedtoidealtriangulations

of punctured surfaces. We will call such representations (quantum) dilogarithmic representations, since the

main ingredient in the theory is the non-compact quantum dilogarithm function ﬁrst introduced in the

context of quantum integrable systems by L.D. Faddeev in [5].

Theserepresentations areinﬁnite dimensional so that a prioriit is not clear if they come from suitable2+1-

dimensionaltopologicalquantumﬁeldtheories(TQFT).Nonetheless,itisexpectedthatinthesingularlimit,

1when the deformation parametertends to a rootof unity , the ”renormalized”theorycorrespondsto a ﬁnite

dimensional TQFT ﬁrst deﬁned in [13, 14] by using the cyclic representations of the Borel Hopf sub-algebra

BU (sl(2)), and subsequently developed and generalized in [1]. One can get the same ﬁnite dimensionalq

representationsofPtolemymodulargroupoidsdirectlyfromcompactrepresentationsofquantumTeichmu¨ller

theory at roots of unity [3, 2, 15].

Projectiverepresentationsofagrouparewellknownto beequivalenttorepresentationsofcentralextensions

of the same group by means of the following procedure. To a group G, aC-vector space V and a group

∗ ∗homomorphism h :G→ PGL(V)≃GL(V)/C , whereC is identiﬁed with a (normal) subgroup of GL(V)

∗ethrough the imbedding z →zid , one can associate a central extension G of G by a sub-group A ofCV

e etogether with a representation h : G→ GL(V) such that the following diagram is commutative and has

exact rows:

∗1 C GL(V) PGL(V) 1

e hh

e1 A G 1G

∗This version: February 2010. L.F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311. R.M.K. is partially

supported by Swiss National Science Foundation.

log(q)1One should distinguish between two diﬀerent limits, depending on whether tends to a positive or a negative rational

2πi

number. In the case when this limit is a positive rational number, the limit of the representation is non-singular and so it stays

inﬁnite dimensional.

1

//OO/?///O////O///O///OeOnesuchextensionis thepull-backG ofthe centralextensionGL(V)→PGL(V) underthe homomorphism

G→PGL(V), which is canonicallydeﬁned. Howeverit is possible to ﬁnd also smaller extensions associated

∗ ∗eto propersub-groupsA⊂C . ThecentralextensionG associatedto the smallestpossiblesub-groupA⊂C

for which there exists a linear representationas in the diagramabove resolving the projectiverepresentation

eof G will be called the minimal reduction of G.

In this light, quantum Teichmu¨ller theory gives rise to representations of certain central extensions of the

surface mapping class groups which are the vertex groups of the Ptolemy modular groupoids. The main

goal of this paper is to identify the isomorphism classes of those central extensions. Namely, by using the

quantization approach of [15], we extend the analysis of the particular case of a once punctured genus three

surface performed in [16] to arbitrary punctured surfaces of ﬁnite type.

Let a group G with a given presentation be identiﬁed as the quotient group F/R, where F is a free group

and R, the normal subgroup generated by the relations. Then, a central extension of G can be obtained

∗from a homomorphism h: F → GL(V) with the property h(R)⊂C so that it induces a homomorphism

h: G→PGL(V). In this case, the homomorphism h will be called an almost linear representation of G, in

order to distinguish it from a projective representation.

In quantum Teichmu¨ller theory, central extensions of surface mapping class groups appear through almost

s slinear representations. Speciﬁcally, let Γ be the mapping class group of a surface Σ of genus g withg,r g,r

r boundary components and s punctures. These are mapping classes of homeomorphisms which ﬁx the

s sboundary point-wise and ﬁx the set of punctures (not necessarily point-wise). Denoting Γ = Γ , theg g,0

sprojective representations of Γ for (2g−2+2s)s > 0, constructed in [15, 16], are almost linear represen-g

s s tetations corresponding to certain central extensions Γ . By considering embeddings Σ ⊂ Σ , the centralg g,r h,0

s seextensions Γ can be used to deﬁne central extensions for the mapping class groups Γ for s≥ r, and theg g,r

associated surfaces containing on each boundary component at least one puncture. According to [21], any

s t t sembedding Σ ⊂Σ , for which Σ \Σ contains no disk or cylinder components, induces an embeddingg,r g,rh,0 h

seof the corresponding mapping class groups. Using this fact, we can deﬁne the central extension Γ as theg,r

t s tepull-back of the central extension Γ by the injective homomorphism Γ ֒→ Γ induced by an embeddingh g,r h

of the correspondingsurfaces. A priori, it is not clear whether such deﬁnition depends on a particular choice

of the embedding, but our main result below shows that this is indeed the case.

Centralextensions by an Abelian groupA of a given groupG areknown to be classiﬁed, up to isomorphism,

2 sby elements of the 2-cohomology group H (G;A). In the case of surface mapping class groups Γ , theg,r

latter was ﬁrst computed by Harer in [12] for g≥ 5 and further completed by Korkmaz and Stipsicz in [18]

for g≥ 4 (see also [17] for a survey). Speciﬁcally, we have

2 s s+1H (Γ )=Z , if g≥ 4,g,r

where the generators are given by (one fourth of) the Meyer signature class χ (it is the only generator for

2 2∼the groups H (Γ )=H (Γ )≃Z, see [20, 12, 18] for its deﬁnition) and s Euler classes e associated withg g,1 i

2 s s+1s punctures. In the case when g =3, the group H (Γ ) still contains the sub-groupZ generated by the3,r

above mentioned classes, but it is not known whether there are other (2-torsion) classes. When g = 2 we

2 s swill show that H (Γ ) contains the subgroupZ/10Z⊕Z , whose torsion part is generated by χ and whose2,r

free part is generated by the Euler classes. The Universal Coeﬃcients Theorem permits then to compute

2H (G;A) for every Abelian group A.

s ∗gsDenote as above by Γ the canonical central extension of Γ byC which is obtained as the pull-back ofg,r g,r

the canonical central extension GL(H)→PGL(H) under the quantum projective representation associated

to a semi-symmetric T in the Hilbert spaceH (see the next section). Quantum representations depend on

∗some parameter ζ∈C . Our main result is the following theorem.

s sgs eTheorem 0.1. The central extension Γ can be reduced to a minimal central extension Γ of Γ by a,r g,r g,r

∗ ∗ −6cyclic Abelian A⊂C , where A is the subgroup of C generated by ζ . Moreover its cohomology class is

sX

2 sc =12χ+ e ∈H (Γ ;A)es i g,rΓg,r

i=1

if g≥ 2 and s≥4. Here χ and e are one fourth of the Meyer signature class and respectively the i-th Euleri

class with A coeﬃcients.

There is a geometric interpretation of this extension.

2\Corollary 0.2. Let us consider the extension Γ of class 12χ. Then there is an exact sequence:g,r+s

s−1 gs\1→A →Γ →Γ →1g,r+s g,r

In some sense the quantum representations of punctured mapping class groups can be lifted to the mapping

class groups of surfaces with boundary obtained by blowing up the punctures.

gsCorollary 0.3. The cohomology class of the central extension Γ is,r

sX

2 s ∗

c =12χ+ e ∈H (Γ ;C )es i g,rΓg,r

i=1

2 s ∗if g≥ 3 and s≥4. The same formula holds also when g = 2 but the class χ vanishes in H (Γ ;C ). Hereg,r

∗χ and e are one fourth of the Meyer signature class and respectively the i-th Euler class withC coeﬃcients.i

Remark 0.1. The central extension arising from SU(2)-TQFT with p -structures was computed in [10, 19]1

for Γ and it equals 12χ. It can be shown that their computations extend to the case of punctured surfacesg Pssand the associated class for Γ is 12χ+ e . Our result shows that this extension coincides with theig,r i=1

central extension arising from quantum Teichmu¨ller theory.

Theorganizationofthepaperis asfollows. InSection1, wereviewthequantizationoftheTeichmu¨llerspace

ofapuncturedsurfaceanddeﬁnetheassociatedquantumrepresentationsofthedecoratedPtolemygroupoid

which correspond to linear representations of a central extension of the decorated Ptolemy groupoid. Then,

in Section 2, we prove Theorem 0.1 by ﬁnding the pull-back of this central extension to the mapping class

groupofthesurface. ThekeyideaistouseaGrothendiecktypeprinciple. Namely,onecanidentifyacentral

extensionofthemapping classgroupofsomesurface,if oneunderstands its restrictionstothe mappingclass

groups of sub-surfaces of bounded topological types. The core of the proof consists in computing explicitly

the lifts to the central extension of the decorated Ptolemy groupoid of the relations known to hold in the

mapping class groups. When properly interpreted, these lifts yield the class of the mapping class group

extension.

Acknowledgements

The authors are indebted to Stephane Baseilhac and Vlad Sergiescu for useful discussions.

1 Quantum Teichmu¨ller theory

1.1 The groupoid of decorated ideal triangulations

sLet Σ = Σ be an oriented closed surface of genus g with s punctures. Denote M = 2g−2+s and assumeg

that Ms> 0. Then surface Σ admits ideal triangulations with vertices at the s punctures.

Deﬁnition 1.1. A decorated ideal triangulation of Σ is an ideal triangulation τ, where all triangles are

provided with a marked corner, and a bijective ordering map

τ¯:{1,...,2M}∋j →τ¯ ∈T(τ)j

is ﬁxed. Here T(τ) is the set of all triangles of τ.

Graphically, the marked cornerof a triangleis indicated by an asteriskand the corresponding number is put

inside the triangle. The set of all decorated ideal triangulations of Σ is denoted Δ .Σ

Recall that if a group G freely acts in a set X, then there is an associated groupoid deﬁned as follows. The

objects are the G-orbits in X, while morphisms are G-orbits in X×X with respect to the diagonal action.

Denote by [x] the object represented by an element x∈X and [x,y] the morphism represented by a pair of

elements (x,y)∈ X×X. Two morphisms [x,y] and [u,v], are composable if and only if [y] = [u] and their

composition is [x,y][u,v] = [x,gv], where g∈ G is the unique element sending u to y. The inverse and the

−1identity morphisms are given respectively by [x,y] = [y,x] and id =[x,x]. In what follows, products of[x]

the form [x ,x ][x ,x ]···[x ,x ] will be shortened as [x ,x ,x ,...,x ,x ].1 2 2 3 n−1 n 1 2 3 n−1 n

The mapping class groupM of Σ acts freely on Δ . In this case, we letG denote the correspondingΣ Σ Σ

groupoid, called the groupoid of decorated ideal triangulations, or decorated Ptolemy groupoid. There is a

presentation forG with three types of generators and four types of relations.Σ

3σ σThe generators are of the form [τ,τ ], [τ,ρ τ], and [τ,ω τ], where τ is obtained from τ by replacing thei i,j

ordering map τ¯ by the map τ¯◦σ, where σ∈ S is a permutation of the set{1,...,2M}, ρ τ is obtained2M i

fromτ by changing the marked corner of triangleτ¯ as in Figure 1, and ω τ is obtained from τ by applyingi i,j

the ﬂip transformation in the quadrilateral composed of triangles τ¯ and τ¯ as in Figure 2.i j

r r

ρ ∗i@ @−→i i

r @∗ r r @r

Figure 1: The transformation ρ .i

r r

@ @

iωi,jr @r r∗ @r∗ i j −→

@ @ j

∗ ∗@r @r

Figure 2: The transformation ω .i,j

There are two sets of relations satisﬁed by these generators. The ﬁrst set is as follows:

α α β αβ

[τ,τ ,(τ ) ]=[τ,τ ], α,β∈S , (1)2M

[τ,ρ τ,ρ ρ τ,ρ ρ ρ τ] = id , (2)i i i i i i [τ]

[τ,ω τ,ω ω τ,ω ω ω τ] = [τ,ω τ,ω ω τ], (3)ij ik ij jk ik ij jk ij jk

(ij) (ij) (ij)[τ,ω τ,ρ ω τ,ω ρ ω τ] = [τ,τ ,ρ τ ,ρ ρ τ ], (4)ij i ij ji i ij j i j

where the ﬁrst two relations are evident, while the other two are shown graphically in Figures 3, 4.

r r r

#c #c #c# c # c # c

# c # c # i c

r# cr r# cr #r∗ cr∗∗

i k ω Z i k ω Zi,j i,kB B B B BZ BZ −→ −→

kZ ZB B B j Z Z

B B B j jZ Z∗ ∗ ∗Br∗ r B∗r Zr B∗r Zr

ω ωj,k j,kց r r ւ

#c #c# c # c

# c # i c

r# cr #r cr∗∗ i j ωi,j B B B B −→

j B B

B B k k ∗ ∗Br ∗r Br ∗r

Figure 3: The Pentagon relation (3).

The following commutation relations fulﬁll the second set of relations:

σ σ σ[τ,ρ τ,(ρ τ) ]=[τ,τ ,ρ −1 τ ], (5)i i σ (i)

σ σ σ[τ,ω τ,(ω τ) ]= [τ,τ ,ω −1 −1 τ ], (6)ij ij σ (i)σ (i)

[τ,ρ τ,ρ ρ τ]= [τ,ρ τ,ρ ρ τ], (7)j i j i j i

[τ,ρ τ,ω ρ τ] = [τ,ω τ,ρ ω τ], i ∈j{,k}, (8)i jk i jk i jk

[τ,ω τ,ω ω τ] =[τ,ω τ,ω ω τ], {i,j}∩{k,l}=∅. (9)ij kl ij kl ij kl

1.2 Hilbert spaces of square integrable functions associated to triangulations

In what follows, we work with Hilbert spaces

2 ⊗n 2 nH≡L (R), H ≡L (R ).

Any two self-adjoint operators p and q, acting inH and satisfying the Heisenberg commutation relation

−1

pq−qp=(2πi) id , (10)H

4r r

@ @

iωi,jr @r r∗ @r∗ i j −→

@ @ j

∗ ∗@r @r

↓(i,j)◦ρ ×ρ ρ↓i j i

r r

@ @iωj,ir @r r @∗ rj i ∗ ←−

@ @ j

∗ ∗@r @r

Figure 4: The Inversion relation (4).

can be realized as diﬀerentiation and multiplication operators. Such ”coordinate” realization in Dirac’s

bra-ket notation has the form

1 ∂

hx|p = hx|, hx|q =xhx|. (11)

2πi∂x

Formally, the set of ”vectors”{|xi} forms a generalized basis ofH with the following orthogonality andx∈R

completeness properties: Z

hx|yi =δ(x−y), |xidxhx| = id .H

R

For any 1≤i≤m we shall use the following notation

⊗mι : EndH∋ a →a = 1⊗···⊗1⊗a⊗1⊗···⊗1∈ EndH .i i | {z }

i−1 times

⊗kBesides that, if u∈EndH for some 1≤k≤m and{i ,i ,...,i}⊂{1,2,...,m}, then we shall write1 2 k

u ≡ι ⊗ι ⊗···⊗ι (u).i i ...i i i i1 2 2 1 2 k

⊗mThe symmetric group S naturally acts inH :m

P (x ⊗···⊗x ⊗···⊗x )=x −1 ⊗···⊗x −1 ⊗...⊗x −1 , σ∈S . (12)σ 1 i m σ (1) σ (i) σ (m) m

1.3 Semi-symmetric T-matrices

We deﬁne now the algebraic structure needed for constructing representations of the decorated Ptolemy

groupoidG .Σ

⊗2Deﬁnition 1.2. A semi-symmetric T-matrix consists of two operators A∈ End(H) and T∈ End(H )

satisfying the equations:

3A = 1, (13)

T T T = T T , (14)12 13 23 23 12

T A T =ζA A P , (15)12 1 21 1 2 (12)

∗where ζ∈C and the permutation operator P is deﬁned by equation (12), for σ denoting the transposition(12)

(12).

Examples of semi-symmetric T-matricescould be obtained as follows. Fix some self-conjugateoperators p,q

satisfying the Heisenberg commutation relation(10). Choose a parameter b satisfying the condition:

(1−|b|)Imb = 0,

and deﬁne then two unitary operators by the following formulas:

2 2−iπ/3 i3πq iπ(p+q)A≡e e e ∈End(H), (16)

i2πp q ⊗21 2T≡e ϕ (q +p −q )∈ End(H ). (17)b 1 2 2

They satisfy the deﬁning relations for a semi-symmetric T-matrix, where

2 iiπc /3 −1bζ =e , c = (b+b ), (18)b

2

5and ϕ is the Faddeev non-compact quantum logarithm deﬁned on{z∈C;|Im(z)| <|Im(c )|} by means ofb b

Z ∞1 exp(−2πizx)dx

ϕ (z)= exp − (19)b

4 sinh(xb)sinh(x/b)x−∞

Its main feature is the following functional equation it satisﬁes:

ϕ (q)ϕ (p) =ϕ (p)ϕ (p+q)ϕ (q)b b b b b

1whenever pq−qp= 1.

2πi

Remark that the operator A is characterized (up to a normalization factor) by the equations:

−1 −1AqA = p−q, ApA =−q.

Note that equations (13)—(15) correspond to relations (2)—(4).

Let us introduce now some notation which will be useful in the sequel. For any operator a∈EndH we set:

−1 −1

a ≡ A a A , a ≡ A a A . (20)ˆ k k ˇ k kkk k k

It is evident that

a = a = a , a = a , a = a ,ˇ ˆ k ˆ ˇ ˇ ˆˆ ˇ k kk k kk

where the last two equations follow from equation (13). In particular, we have

p =−q , q = p −q , (21)ˆ k ˆ k kk k

p = q −p , q =−p . (22)ˇ k k ˇ kk k

Besides that, it will be also useful to use the notation

−1P ≡ A P , P ≡ A P , (23)ˇˆ k (kl...m) (kl...m)(kl...mk) k(kl...mk)

where (kl...m) is the cyclic permutation

(kl...m): k →l →... →m →k.

Equation (15) in this notation takes a rather compact form

T T =ζP . (24)12 ˆ ˆ21 (121)

Remark 1.1. One can derive the following symmetry property of the T-matrix: T = T .12 ˆˇ21

1.4 The quantum Teichmu¨ller space

The quantization of the Teichmu¨ller space of a punctured surface Σ induced by a semi-symmetric T-matrix

is deﬁned by means of a quantum functor:

⊗2MF:G →End(H ),Σ

Its meaning is that we have a operator valued function:

⊗2MF: Δ ×Δ →End(H ),Σ Σ

satisfying the following equations:

′ ′ ′′ ′′ ′ ′′F(τ,τ) = id ⊗2M, F(τ,τ )F(τ ,τ )F(τ ,τ)∈C\{0}, ∀τ,τ ,τ ∈Δ , (25)H Σ

′ ′F(f(τ),f(τ )) = F(τ,τ ), ∀f∈M , (26)Σ

F(τ,ρ τ)≡ A , (27)i i

F(τ,ω τ)≡ T , (28)i,j ij

σF(τ,τ )≡ P , ∀σ∈S , (29)σ 2M

where operator P is deﬁned by equation (12). Consistency of these equations is ensured by the consistencyσ

of equations (13)—(15) with relations (2)—(4).

6′′A particular case of equation (25) corresponds to τ =τ:

′ ′

F(τ,τ )F(τ ,τ)∈C\{0}. (30)

−1 −1′As an example, we can calculate the operator F(τ,ω (τ)). Denoting τ ≡ω (τ) and using equation (30),i,j i,j

as well as deﬁnition (28), we obtain

−1 ′ ′ ′ ′ −1 −1F(τ,ω (τ)) = F(ω (τ ),τ )≃ (F(τ ,ω (τ ))) = T , (31)i,j i,ji,j ij

where≃ means equality up to a numerical multiplicative factor.

The quantumfunctor induces a unitaryprojectiverepresentationof the mapping class groupM as follows:Σ

⊗2MM ∋f →F(τ,f(τ))∈ End(H ).Σ

Indeed, we have the following relation (up to a non-zero scalar):

F(τ,f(τ))F(τ,h(τ)) = F(τ,f(τ))F(f(τ),f(h(τ)))≃ F(τ,fh(τ)).

The main question addressed in this present paper is to identify the central extension of the mapping class

groupcorrespondingtothisprojectiverepresentation. Observethattheprojectivefactorliesinthesub-group

∗ofC generated by ζ.

se2 Presentation of Γg,r

2.1 Generating set for the relations

0We start with a number of notations and deﬁnitions. Let us denote Σ = Σ .g,r g,r

s sDeﬁnition 2.1. A chain relation C on the surface Σ is given by an embedding Σ ⊂ Σ and the1,2g,r g,r

standard chain relation on this 2-holed torus, namely

4(D D D ) =D Da b c e f

where a,b,c,d,e,f are the following curves of the embedded 2-holed torus:

a

be f

c

s sDeﬁnition 2.2. A lantern relation L on the surface Σ is given by an embedding Σ ⊂ Σ and the0,4g,r g,r

standard lantern relation on this 4-holed sphere, namely

−1 −1 −1 −1D D D D D D D = 1 (32)a a a12 13 23 a a a a0 1 2 3

where a ,a ,a ,a ,a ,a ,a are the following curves of the embedded 4-holed sphere:0 1 2 3 12 13 23

a a212 a

23

a0

a a1 3

a13

1 s 1Deﬁnition 2.3. Consider an embedding Σ ⊂ Σ such that the boundary components a ,a ,a of Σ1 2 30,3 g,r 0,3

1are non-separating curves. Let then a ,a ,a be embedded curves on Σ so that a bounds a pair of12 13 23 jk0,3

1pants Σ ⊂ Σ along with a and a , for all 1≤ j = k≤ 3. Then the puncture relation P (supported at0,3 j k0,3

1 sthe puncture of Σ ) on the surface Σ is:0,3 g,r

−1 −1 −1D D D D D D = 1 (33)a a a12 13 23 a a a1 2 3

7

6Remark 2.1. The puncture relation is, in fact, a consequence of the lantern relation and the fact that the

Dehn twist along a small loop encircling a puncture is trivial.

seThe ﬁrst step in proving Theorem 0.1 is to ﬁnd an explicit presentation for the central extension Γ .g,r

Speciﬁcally, by using Gervais’ presentation [10], we have the following description.

seProposition 2.1. Suppose that g≥ 2 and s≥4. Then the group Γ has the following presentation.g,r

1. Generators:

s e(a) With each non-separating simple closed curve a in Σ is associated a generator D ;ag,r

(b) One (central) element z.

2. Relations:

(a) Centrality:

e ezD =D z (34)a a

sfor any non-separating simple closed curve a on Σ ;g,r

(b) Braid type 0-relations:

e e e eD D =D D (35)a b b a

for each pair of disjoint non-separating simple closed curves a and b;

(c) Braid type 1-relations:

e e e e e eD D D =D D D (36)a b a b a b

for each pair of non-separating simple closed curves a and b which intersect transversely at one

point;

(d) One lantern relation on a 4-holed sphere subsurface with non-separating boundary curves:

e e e e e e eD D D D =D D D (37)a a a a a a a0 1 2 3 12 13 23

(e) One chain relation on a 2-holed torus subsurface with non-separating boundary curves:

4 12e e e e e(D D D ) =z D D (38)a b c e f

(f) Puncture relations:

e e e e e eD D D =zD D D (39)a a (i) a (i) a (i) a (i) a (i)12(i) 13 23 1 2 3

sfor each puncture p of Σ , i∈{1,2,...,s}.i g,r

(g) Scalar equation:

Nz = 1 (40)

−6 ∗where N is the order of ζ , in the case where ζ∈C is a root of unity.

2.2 Proof of Proposition 2.1

e e e e eLemma 2.1. For any lifts D of the Dehn twists D we have D D = D D and thus relations (2) area a a b b a

satisﬁed.

Proof. Thecommutativityrelationsaresatisﬁedforparticularlifts comingfromasemi-symmetricT-matrix.

If we changethe lifts by multiplying each lift by some central element the commutativity is still valid. Thus,

the commutativity holds for any lifts.

eLemma 2.2. There are lifts D of the Dehn twists D , for each non-separating simple closed curve a sucha a

e e e e e ethat we have D D D =D D D for any simple closed curves a,b with one intersection point, and thus thea b a b a b

braid type 1-relations (3) are satisﬁed.

8Proof. Consider an arbitrary lift of one braid type 1-relation (to be called the fundamental one), which has

k ke e e e e e e ethe form D D D = z D D D . Change then the lift D into z D . With the new lift the relation abovea b a b a b b b

e e e e e ebecomes D D D =D D D .a b a b a b

sChoose now an arbitrary braid type 1-relation of Γ , say D D D = D D D . There exists a 1-holedx y x y x yg,r

storus Σ ⊂Σ containingx,y, namely a neighborhoodofx∪y. LetT be the similar torus containinga,b.1,1 g,r

s sSince a,b and x,y are non-separating there exists a homeomorphism ϕ : Σ → Σ such that ϕ(a) = xg,r g,r

and ϕ(b) =y. We have then

−1 −1D =ϕD ϕ , D =ϕD ϕ .x a y b

Let us consider now an arbitrary lift ϕe of ϕ, which is well-deﬁned only up to a central element, and set

−1 −1e e e eD =ϕeD ϕe , D =ϕeD ϕe .x a y b

These lifts are well-deﬁned since they do not depend on the choice of ϕe (the central elements coming from

−1ϕe and ϕe mutually cancel). Moreover, we have then

e e e e e eD D D =D D Dx y x y x y

and so the braid type 1-relations (3) are all satisﬁed.

eLemma 2.3. The choice of lifts of all D , with x non-separating, satisfying the requirements of Lemma 2.2x

eis uniquely deﬁned by ﬁxing the lift D of one particular Dehn twist.a

se eProof. In fact the choice of D ﬁxes the choice of D . If x is a non-separating simple closed curve on Σ ,a b g,r

then there exists another non-separating curve y which intersects it in one point. Thus, by Lemma 2.2, the

echoice of D is unique.x

seLemma 2.4. One can choose the lifts of Dehn twists in Γ so that all braid type relations are satisﬁed andg,r

the lift of the lantern relation is trivial, namely

e e e e e e eD D D D =D D Da b c d u v w

sfor the non-separating curves on an embedded Σ ⊂Σ .0,4 g,r

ke e e e e e eProof. ArbitraryliftofthatlanternrelationisoftheformD D D D =z D D D . Inthiscase,wechangea b c d u v w

−ke ethe lift D into z D and adjust the lifts of all other Dehn twists along non-separatingcurves the waythata a

all braid type 1-relations are satisﬁed. Then, the required form of the lantern relation is satisﬁed.

We say that the lifts of the Dehn twists are normalized if all braid type relations and one lantern relation

are lifted in a trivial way.

Lemma 2.5. A normalized Dehn twist in quantum Teichmu¨ller theory is conjugated to the inverse T-matrix

−6times ζ i.e.

−6 −1 −1eD = F(τ,D τ) =ζ U T U .α α α αkl

Proof. The idea of the proof is to calculate the lift of the lantern relation. Consider the following decorated

triangulation τ of the 4-holed disk with 4 punctures:

a0

a 12

4

a 21

5 3a 63

7

8

The trick used in [15, 16] for computing D is to use a sequence of ﬂips to change the triangulation into onea

which intersects some curve isotopic to a into two points. Then the Dehn twist along a can be expressed

as the ﬂip of one of the two edges of the latter triangulation intersecting a. This recipe generalizes to the

case where the curve a intersects several edges of the triangulation, if a is a boundary component with

one puncture on it. Speciﬁcally, let e ,...,e be the edges issued from the puncture, in counterclockwise1 s

order. Then the Dehn twist D can be expressed as the result of composing the ﬂips of e ,e ,...,e . Wea 1 2 s−1

−1illustrate this procedure with the case of the left Dehn twist D on the triangulation τ above:a3

9

1 1

4 4T5 3

2 2

5 3 6 35

6

7 788

T 3 8

1

4

2

83

5 6

7

T

3 7

1 1

4 4

T

2 3 6 2

8 8

55 63 7 367

In particular, we ﬁnd that the following expression for the right Dehn twist along a :3

¯ ¯F = F(τ,D τ) =T T T T (41)ˇa a ˆ ˆ ˆ3 3 35 38 37 36

We use further the same recipe for the remaining Dehn twists along boundary components and obtain:

¯ ¯F = F(τ,D τ) =T T T T (42)ˇa a 24 25 261 1 23

¯ ¯F = F(τ,D τ) =T T T T (43)ˇ ˇa2 a2 ˆ 1714 12 16

¯ ¯F = F(τ,D τ) =T T T T (44)ˇˇ ˇˇ ˇˇa0 a0 ˇˆ 84 81 8785

In order to compute T we need to transform the triangulation τ into one which intersects a curvea12

isotopic to a into precisely two points. This can be done as follows:12

10