Wolpert formulae for the Poisson bracket of Hitchin Length Functions November 17, 2016 Abstract In this paper, we use Goldman’s formula for the Poisson bracket of invariant functions to derive formulae for the Poisson bracket of length functions on Hitchin space. In particular we derive the formulae for the first and second derivatives {l i α ,l j β } and {{l i α ,l j β },l k γ }. These generalize formulae of Wolpert given in the Teichm¨ uller case for the Weil-Petersson Poisson bracket. We use the formula to prove convexity results for spectral length functions along the associated Hamiltonian vector fields. We also derive a formula for H{l i α ,l j β }∩ [π], the Poincare dual of the Hamiltonian of the Poisson bracket of two length functions. Contents 1 Introduction 2 2 Wolpert’s formulae and their generalizations 3 3 The Generalized Wolpert Formulae 4 3.1 Cosine Formula for {l i α ,l j β } ................................ 4 3.2 Sine Formula for {{l i α ,l j β },l k γ } ............................... 6 4 Goldman’s Symplectic Form 8 5 Proof of Generalized Wolpert Cosine formula 9 5.1 Eigenvalue Perturbation .................................. 9 5.2 Poission bracket of length functions ............................ 11 6 Proof of Generalized Wolpert Sine Formula 12 6.1 Differential of Projection maps .............................. 12 6.2 Homology with coefficients in a flat bundle ....................... 14 1
Transcript
Wolpert formulae for the Poisson bracket of Hitchin Length
Functions
November 17, 2016
Abstract
In this paper, we use Goldman’s formula for the Poisson bracket of invariant functions toderive formulae for the Poisson bracket of length functions on Hitchin space. In particularwe derive the formulae for the first and second derivatives {liα, l
jβ} and {{liα, l
jβ}, lkγ}. These
generalize formulae of Wolpert given in the Teichmuller case for the Weil-Petersson Poissonbracket. We use the formula to prove convexity results for spectral length functions along theassociated Hamiltonian vector fields. We also derive a formula for H{liα, l
jβ} ∩ [π], the Poincare
dual of the Hamiltonian of the Poisson bracket of two length functions.
We let S be a closed oriented surface of genus g ≥ 2, G a semisimple Lie group and define
R(S,G) = Hom(π1(S),G)/G
the space of conjugacy classes of representations.
For G = PSL(2,R), Goldman proved that R(S,G) has exactly 4g − 3 connected components,with two components corresponding to the Teichmuller spaces T (S) and T (S) of discrete-faithfulrepresentations (see [5]).
For G = PSL(n,R) Hitchin studied the connected components of R(S,G) containing the image ofa Teichmuller component of R(S,PSL(2,R)) under the irreducible representation from PSL(2,R)to PSL(n,R) (see Hitchin’s original paper [7]). These components are called Hitchin componentsand denoted Hn(S), Hn(S). In particular for n = 2, there are two Hitchin components given byH2(S) = T (S) and H2(S) = T (S). Hitchin components can be considered as a generalization ofTeichmuller space to representations into PSL(n,R). In particular, Hitchin proved;
Theorem 1 (Hitchin, [7]) For n odd there is one Hitchin component Hn(S) and for n eventhere are two Hn(S), Hn(S). Each Hitchin component is an analytic manifold diffeomorphic toR|χ(S)|(n2−1).
Using dynamical methods, Labourie showed that Hitchin representations have a rich geometricstructure. In particular Labourie proved the following;
Theorem 2 (Labourie, [9]) If ρ is a Hitchin representation then ρ is discrete faithful and for everyg 6= e, ρ(g) is diagonalizable over R with eigenvalues distinct λ1(g), . . . , λn(g) satisfying
|λ1(g)| > |λ2(g)| > . . . > |λn(g)|.
Given α a free homotopy class of a closed oriented curve in S, we therefore have functions liα :Hn(S)→ R given by
liα([ρ]) = log |λi(ρ(α))|.
We also define the width function
Wα = l1α − lnα = log
∣∣∣∣λ1(α)
λn(α)
∣∣∣∣which describes the spectral width of α.
2
In [1], Atiyah-Bott introduced a symplectic structure on the unitary representation variety of asurface group. Goldman generalized this to define a symplectic structure ω on the representationvariety of a surface group into a general reductive group (see [6]). This gives a symplectic formω on Hn(S) and an associated Poisson structure on the space of smooth functions on Hn(S). Forn = 2, Goldman showed that ω = 2ωwp where ωwp is the Weil-Petersson symplectic form.
2 Wolpert’s formulae and their generalizations
We now describe Wolpert’s Cosine and Sine formulae for length functions on T (S). For α ∈ π1(S)we let lα : T (S)→ R be the length function for α.
We will also state our main results and prior results generalizing Wolpert’s formulae. We will leaveoff until the next section a full description of the various terms that arise in these generalizations.
Wolpert proved the following;
Theorem 3 (Wolpert’s Cosine Formula, [14]) Let α, β be homotopy classes of closed orientedcurves in S with unique closed geodesic representatives α′, β′ in S, then
{lα, lβ}wp =∑
p∈α′∩β′
cos θp
where θp ∈ (0, π) is the angle of intersection from α′ to β′ at p (measured counterclockwise).
As part of his proof of the Nielson realization problem, Kerckhoff proved the above formula in thecase of α, β being measured laminations (see [8]).
Labourie generalized this for the spectral width functions on Hn(S) for simple curves intersectingonce. Specifically he proved
Theorem 4 (Labourie, [11]) Let α, β be homotopy classes of simple closed curves in S intersectingat most once represented by simple closed curves α, β in S in general position. Let α∩ β = p, then
{Wα,Wβ}([ρ]) = ε(p, α, β)∑
ε,ε′∈{1,−1}
ε.ε′b(ρp([αp])ε, ρp([βp])
ε′)
where b is the cross-ratio, ρp ∈ [ρ] is a choice of representation such that ρp : π1(S, p)→ PSL(n,R)and αp, βp are the curves α, β based at p.
Our first result is to prove a general Wolpert Cosine formula for all curves and length functions.We will prove;
Theorem 5 Let α, β be homotopy classes of closed oriented curves in S represented by immersedcurves α, β in S which are in general position, then
{liα, ljβ}([ρ]) =
∑p∈α∩β
εp(α, β)
(bij(ρp([αp]), ρp([βp]))−
1
n
).
where bij is a cross-ratio, ρp ∈ [ρ] is a choice of representation such that ρp : π1(S, p)→ PSL(n,R)and αp, βp are the curves α, β based at p.
3
Wolpert also derived the following Sine formula for the Poisson triple bracket on T (S). This formulawas important in proving the convexity of length functions along Weil-Peterson geodesics.
Theorem 6 (Wolpert’s Sine Formula, [14]) Let α, β, γ be homotopy classes of closed orientedcurves in S with unique closed geodesic representatives α′, β′, γ′ in S, then
{{lα, lβ}wp, lγ}wp =∑
(p,q)∈α∩β×α∩γ
elpq + elqp
2(elα − 1)sin θp sin θq −
∑(p,r)∈β∩α×β∩γ
empr + emrp
2(elβ − 1)sin θp sin θr
where θx is the angle at the given intersection point and lxy is the length along α between x, y(similarly mxy is the length along β.)
Recently, Long derived a generalization of Wolpert’s Sine formula for the case of H3(S) using twist-bulge deformations (see [12]). We will derive a formula for the triple bracket in the general case.We prove;
Theorem 7 Let α, β, γ be homotopy classes of closed oriented curves in S represented by immersedcurves α, β, γ in S which are in general position, then
{{liα, ljβ}, l
kγ} =∑
l 6=i
∑(p,q)∈α∩β×α∩γ
εp(α, β)εq(α, γ)
(λl(α)
λi(α)− λl(α)bijlkρq (αq, βpq, αq, γq) +
λi(α)
λi(α)− λl(α)bljikρq (αq, βpq, αq, γq)
)−∑
l 6=j
∑(p,r)∈β∩α×β∩γ
εp(β, α)εr(β, γ)
(λl(β)
λj(β)− λl(β)bjilkρr (βr, αpr, βr, γr) +
λj(β)
λj(β)− λl(β)blijkρr (βr, αpr, βr, γr)
)
We will describe all the above terms in the following sections.
3 The Generalized Wolpert Formulae
3.1 Cosine Formula for {liα, ljβ}
Following the work of Labourie (see [10]), we consider the cross-ratio on lines and planes. ForV a vector space, we let P (V ) be the space of lines in V . We have RPn−1 = P (Rn) and defineRPn−1∗ = P (Rn∗). The cross-ratio is then the map b : RPn−1 × RPn−1∗ × RPn−1 × RPn−1∗ → Rwhere
where x′ ∈ x, y′ ∈ y, z′ ∈ z, w′ ∈ w are any choice of non-zero elements. This is independent of thechoices made.
4
We define Hyp ⊆ SL(n,R) to be the set of matrices which is diagonalizable over R with eigenvalueshaving distinct absolute values. If A ∈ Hyp we order the eigenvalues of A such that
|λ1(A)| > |λ2(A)| > . . . > |λn(A)|.
We let V i(A) ∈ RPn−1 be the λi(A) eigenspace and let ξi(A) ∈ RPn−1∗ be the line of linearfunctionals projecting onto V i(A) parallel to
⊕j 6=i Vj . We define
bij(A,B) = b(V i(A), ξi(A), V j(B), ξj(B)).
If ρ : π1(S)→ PSL(n,R) is a Hitchin representation, and α, β ∈ π1(S) then we define
bijρ (α, β) = bij(ρ(α), ρ(β)).
For two oriented curves α, β intersecting at a point p on an oriented surface, we let εp(α, β) = ±1depending on if the ordered pair of vectors (α, β) define a positively or negatively oriented basisfor the tangent space at p.
We note for α, β homotopy classes of closed oriented curves in S we can choose α, β immersedclosed curves in S in the homotopy classes of α, β respectively. Furthermore we can choose α, β sothat they intersect transversely. If p ∈ α∩β we let αp, βp ∈ π1(S, p) be the homotopy classes given
by traversing α, β starting at p.
Furthermore for and [ρ] ∈ R(S,PSL(n,R)), given any p ∈ S we can choose a ρp : π!(S, p) →PSL(n,R) be such that ρp ∈ [ρ].
For α a homotopy class of closed curve, we have the associated length functions liα : Hn(S) → Rgiven by liα([ρ]) = log |λi(ρ(α))|. Similarly we define the width function Wα = l1α − lnα.
As described above, in [11] Labourie introduced a new Poisson algebra called the swapping algebraand used it to derive a generalization of Wolpert’s cosine formula which we now restate.
Theorem 4 (Labourie, [11]) Let α, β be homotopy classes of simple closed curves in S intersectingat most once represented by simple closed curves α, β in S in general position. Let α∩ β = p, then
{Wα,Wβ}([ρ]) = εp(α, β)∑
ε,ε′∈{1,−1}
ε.ε′b(ρp([αp])ε, ρp([βp])
ε′)
We will now restate our generalization.
Theorem 5 Let α, β be homotopy classes of closed oriented curves in S represented by immersedcurves α, β in S which are in general position, then
{liα, ljβ}([ρ]) =
∑p∈α∩β
εp(α, β)
(bijρp(αp, βp)−
1
n
).
It is easy to check that this recovers the formula of Labourie above.
We now check that this gives Wolpert’s formula for n = 2. In this case there is a single cross-ratio b.Let α, β ∈ PSL(2,R) be hyperbolic with axes intersecting in point p. We conjugate p to the center
5
of the Poincare model, we can take α to have ordered endpoints (−1, 1) and β to have orderedendpoints (e−iφp , eiφp) for some 0 < φp < 2π. Then b(α, β) = cos2(φp/2) giving
b(α, β)− 1
2=
1
2(2 cos2(φp/2)− 1) =
1
2cosφp.
To compare with Wolpert, we note that Wolpert defines θp to be the angle in (0, π) going counter-clockwise from A to B. If φp < π then θp = φp and if φp > π then θp = φp − π. Therefore
b(α, β)− 1
2=
1
2cosφp =
1
2εp(α, β) cos θp. (1)
Now applying the formula in Theorem 5 we have for α, β ∈ π1(S) then
{l1α, l1β} =1
2
∑p∈α∩β
εp(α, β) cosφp =1
2
∑p∈α∩β
εp(α, β)2 cos θp =1
2
∑p∈α∩β
cos θp
As lα = 2l1α we have
{lα, lβ} = 2.∑p∈α∩β
cos θp
As ω = 2.ωwp we obtain the original cosine formula for the Weil-Petersson Poisson bracket.
3.2 Sine Formula for {{liα, ljβ}, lkγ}
To state the generalized Wolpert sine formula we define the following. Given A ∈ Hyp, we can choosea basis of eigenvectors vi(A) ∈ V i(A) and dual linear functionals ξi(A) such that < ξi(A)|vj(A) >=δij . Then we define the projection map P i(A) : Rn → Rn given by
As above, if α, β ∈ π1(S, x) and ρ : π1(S, x)→ PSL(n,R) we define M ijρ (α, β) = M ij(ρ(α), ρ(β)).
We let ∩[π] : H1(π, g) → H1(π, g) be the map X → X ∩ [π] given by cap product with thefundamental class [π] ∈ H2(π,R). Also let H be the Hamiltonian operator with respect to theGoldman symplectic form (see Goldman [4] for details). Then
Theorem 8 Let α, β be homotopy classes of closed oriented curves in S represented by immersedcurves α, β in S which are in general position, then
(H{liα, ljβ} ∩ [π])[ρ] =
∑p∈α∩β
εp(α, β)(α⊗M ij
ρp(αp, βp) + β ⊗M jiρp(βp, αp)
).
where ρp : π1(S, p)→ PSL(n,R) is chosen such that ρp ∈ [ρ].
We now consider {{liα, ljβ}, l
kβ}. We choose curves α, β, γ in the homotopy classes α, β, γ such that
they are immersed curves who intersect pairwise transversely.
For p ∈ α∩ β and q ∈ α∩ γ we have the αpq be the oriented arc of α joining p to q. Given a closedcurve cp based at p, we obtain a closed curve based at q by conjugating by αpq to obtain cpq =α−1pq cpαpq. If hpq : π1(S, p)→ π1(S, q) is the associated isomorphism we have hpq[αp] = [αpq] = [αq].The curve αpq i does not satisfy this property (see Figure 1).
Similarly for p ∈ α ∩ β and r ∈ β ∩ γ we let βpr be the oriented arc on β joining p and r and we
define αpr and βpr by conjugating by βpr. If hpr : π1(S, p)→ π1(S, r) is the associated isomorphismwe have hpr([βp] = [βpr] = [βr].
We now restate our generalized Wolpert Sine formula for the triple Poisson bracket.
Theorem 7 Let α, β, γ be homotopy classes of closed oriented curves in S represented by immersedcurves α, β, γ in S which are in general position, then
{{liα, ljβ}, l
kγ} =
7
∑l 6=i
∑(p,q)∈α∩β×α∩γ
εp(α, β)εq(α, γ)
(λl(α)
λi(α)− λl(α)bijlkρq (αq, βpq, αq, γq) +
λi(α)
λi(α)− λl(α)bljikρq (αq, βpq, αq, γq)
)−∑
l 6=j
∑(p,r)∈β∩α×β∩γ
εp(β, α)εr(β, γ)
(λl(β)
λj(β)− λl(β)bjilkρr (βr, αpr, βr, γr) +
λj(β)
λj(β)− λl(β)blijkρr (βr, αpr, βr, γr)
)This formula was derived by Long in the case of H3(S) using twist-bulge deformations for realconvex projective structures (see [12]).
4 Goldman’s Symplectic Form
We now describe Goldman’s symplectic form and his formula for the Poisson bracket of invariantfunctions as given in [4]. Let G be a reductive matrix group and consider the non-degeneratesymmetric form B : g × g → R given by B(X,Y ) = Tr(XY ). An invariant function for G is afunction f : G → R which is conjugacy invariant. In particular f = Tr is an invariant function.Given f there is a natural function F : G→ g given by
B(F (A), X) =d
dtf(Aexp(tX)) for all X ∈ g
Thus F (A) is dual to L∗A(df(A)) ∈ g∗ under the isomorphism B : g → g∗ given by B(X)(Y ) =B(X,Y ).
Let S be a closed oriented surface of genus g ≥ 2 and π = π1(S, x). We consider the spaceHom(π,G)/G of representations ρ : π → G up to conjugacy and let R(S,G) be the space ofsmooth points of Hom(π,G)/G. If α is a non-trivial homotopy class of closed oriented curve inS then α defines a conjugacy class in π. If f is an invariant function for G then we can definefα : R(S,G)→ R by
fα([ρ]) = f(ρ(α))
where α ∈ α.
The tangent space at [ρ] ∈ R(S,G) can be identified with the group cohomology H1(π, gAd◦ρ).Using B to pair coefficients we have the maps
H1(π, gAd◦ρ)×H1(π, gAd◦ρ)B(.∪.)−−−−→ H2(π,R)
∩[π]−−→ H0(π,R) = R
This map gives the Goldman symplectic form ω on R(S,G)
ω[ρ](X,Y ) = B(X ∪ Y ) ∩ [π].
Given a smooth function f : R(S,G) → R the Hamiltonian vector field of f is the vector fieldHf defined by ω(Hf, Y ) = df(Y ). For f, g two smooth functions the associated Poisson bracketon smooth functions is the pairing {., .} : C∞(R(S,G),R) × C∞(R(S,G),R) → C∞(R(S,G)),R)given by
{f, g}([ρ]) = ω[ρ](Hf,Hg).
8
Given α an oriented curve in S, if p ∈ α, we let αp be the oriented curve given by traversingα starting at p. If α, β are two oriented closed curves, then α, β are in general position if theirintersections are transverse. If α, β are in general position, then for p ∈ α∩β we define εp(α, β) = ±1given by if the orientation of the point of intersection agrees or not with the orientation of thesurface.
Also for [ρ] ∈ R(S,G) we let ρp : π1(S, p)→ G be a representation defined by change of base pointof ρ. This is well-defined up to conjugacy.
Goldman gave the following description of the Poisson bracket for invariant functions.
Theorem 9 (Goldman, [6]) Let f, f ′ : G → R be invariant functions for G with associated func-tions F, F ′ : G→ g. Let α, β be homotopy classes of closed oriented curves represented by immersedcurves α, β in S which are in general position. Then
{fα, f ′β}[ρ] =∑p∈α∩β
εp(α, β)B(F (ρp(αp)), F′(ρp(βp))
5 Proof of Generalized Wolpert Cosine formula
We define the hyperbolic elements Hyp ⊆ PSL(n,R) to be subset which are diagonalizable over Rwith eigenvalues having distinct absolute values. ForA ∈ Hyp, A has eigenvalues±(λ1(A), . . . , λn(A))with |λ1(A)| > |λ2(A)| > . . . > |λn(A)|. The set Hyp ⊆ PSL(n,R) is an open subset.
In order to use Goldman’s formula above, we need to define a length function li : PSL(n,R) → Rwhich will give the length functions liα on Hn(S). Given A ∈ PSL(n,R) if A is diagonalizable inHyp we let Li(A) = log |λi(A)| and if A is not we let Li(A) = 0. Then Li is an invariant functionand furthermore li is analytic on Hyp. We then have that the map liα : Hn(S) → R is given byliα([ρ]) = Li(ρ(α)) which is analytic by the analyticity of Li on Hyp.
We then define the function Li : Hyp→ sln by
B(Li(A), X) =d
dtli(A exp(tX)).
5.1 Eigenvalue Perturbation
We now consider perturbation of eigenvalues in the space of hyperbolic matrices. Given A ∈ Hyplet P i(A) : Rn → R be projection onto the i-th eigenspace, parallel to the other eigenvectors.
Lemma 1 The length function li : Hyp→ R satisfies
dliA(X) =1
λi(A)Tr(P i(A).X).
Proof: We let A = A0 and denote the eigenvalues and eigenvectors of A by λi, xi. We further letA = A0. We have At has eigenvalues λit and eigenvectors xit. We have
At.xit = λit.x
it.
9
Differentiating we getAxi +Axi = λi.xi + λi.xi
We apply the projection map P i(A) to the above equation to get
P i(A)Axi + P i(A)Axi = λi.P i(A)xi + λi.xi
As P i(A)A = λi.P i(A) we have P i(A)Axi = λi.P i(A)xi so after cancellation we get
λi.xi = P i(A)A.xi.
Therefore we haveλi = Tr(P i(A)A).
As li(X) = log |λi(X)| on Hyp we have
dli =dλi
λi.
Therefore
dliA(X) =1
λi(A)Tr(P i(A)X)
�
We now use the above lemma to calculate Li.
Lemma 2
Li(A) = P i(A)− 1
nI.
Proof: As Hyp is open, we have for any TAHyp ' TAPSL(n,R) ' sln. Therefore for any X ∈ sln
B(Li(A), X) =d
dtli(A exp(tX)) = dliA(AX) =
1
λi(A)Tr(P i(A)AX).
By definition P i(A)A = λi(A)P i(A). Therefore
B(Li(A), X) =1
λi(A)Tr(λi(A).P i(A).X) = Tr(P i(A).X).
We let B : gl(n,R) × gl(n,R) → R given by B(X,Y ) = Tr(XY ). Then B is non-degenerate andrestricts to B on g. We let p : gl(n,R) → sln be orthogonal projection with respect to B. Thengiven A ∈ gl(n,R), then for all X ∈ g
We now use Goldman’s formula to prove the generalization of Wolpert’s cosine formula.
Theorem 5
{liα, ljβ}([ρ]) =
∑p∈α∩β
εp(α, β)
(bijρp(αp, βp)−
1
n
).
Proof: From the above we have
B(Li(A), Lj(B)) = Tr
((P i(A)− 1
n.I
).
(P j(B)− 1
n.I
))As Tr(P i(A)) = Tr(P j(B)) = 1
B(Li(A), Lj(B)) = Tr(P i(A)P j(B))− 1
n
Now applying Goldman’s formula from theorem 9 we get
{liα, ljβ}([ρ]) =
∑p∈α∩β
εp(α, β)B(Li(ρp(αp)), Lj(ρp(βp)))
=∑p∈α∩β
εp(α, β)
(Tr(P i(ρp(αp))P
j(ρp(βp))−1
n
).
For any X ∈ Hyp and let V i(X) be the eigenlines and ξi(X) the dual eigenplanes. We let A,B ∈ Hypand we choose non-zero elements ai ∈ V i(A), ai ∈ ξi(A), bj ∈ V j(B), bj ∈ ξj(B). Then
P i(A)(v) =< ai|v >< ai|ai >
ai P j(B)(v) =< bj |v >< bj |bj >
bj .
Similarly for B ∈ Hyp with b+i , b−i . Then if A,B ∈ Hyp we have
We now want to calculate a formula for {{liα, ljβ}, l
kγ}. This will generalize Wolpert’s Sine formula
for second derivatives of length functions.
We have that the tangent space T[ρ]R(S,G) ' H1(π1(S), gAd◦ρ) where Ad ◦ ρ is composition of theAdjoint representation and ρ (see [6]). The identification can be described easily as follows; Let ρtbe a smooth path of representations with ρ0 = ρ then we define the 1-cochain ξ : π1(S) → g fortangent vector ρ0 by
ξ(g) =d
dt
∣∣∣∣t=0
(ρt(g)ρ0(g)−1) ∈ TeG ' g.
A simple calculation show this is closed and that it is exact if it corresponds to conjugation of ρ0.Then the tangent vector [ρ0] corresponds to [ξ] ∈ H1(π1(S), gAd◦ρ).
6.1 Differential of Projection maps
For A ∈ Hyp we label the eigenvalues λi(A) in descending absolute value. We let P i(A) be theprojection onto the λi(A)-eigenspace parallel to the hyperplane spanned by the others. For a choiceof eigenvectors vi(A), we let ξi(A) be the dual basis. We then define maps P ij (A) : Rn → Rn by
P ij (A)(v) =< ξi(A)|v > vj(A).
For the function li : Hyp → R, by Lemma 2 we have the associated invariant function Li :SL(n,R)→ sln is
Li(A) = P i(A)− 1
nI.
Given any linear map Q, we let Qij =< ξi(A)|Qvj(A) > its matrix coefficients with respect to thebasis vi(A).
Further, if ρ : π1(S)→ Hyp ⊂ SL(n,R) is a representation, we define P ij (α) = P ij (ρ(α)).
Lemma 3 If X ∈ T 1[ρ](Hn(S)) = H1(π, slnAdρ) then
dP i(X)(α) =∑j 6=i
λj(α)
λi(α)− λj(α)Xji (α)P ij (α) +
λi(α)
λi(α)− λj(α)Xij(α)P ji (α).
Proof: We let At = ρt(α) and let A = A0, we choose eigenvalues vi and dual eigenplanes ξi suchthat < ξi|vi >= 1 Then as before
We note that a linear map X has matrix coefficients Xij =< ξi|Xvj > with respect to this basis.
Then as ξj ◦A = λjξj we have
Aji = λi.δji + (λi − λj). < ξj |vi > .
For i = j we obtain the formula
λi = Aii =< ξi|Avi >= Tr(AP i(A)) = Tr(P i(A)A).
We define cji =< ξj |vi >. Then for i 6= j
cji =< ξj |vi >=1
λi − λjAji
As < ξj |vi >= δji differentiating we have
< ξj |vi > + < ξj |vi >= 0
Writing vi, ξi in terms of the basis, we therefore have
vi =∑j
cjivj ξi = −∑j
cijξj .
We now writeP i(A) =< ξi|· > vi
Differentiating we get
dP i =< ξi|· > vi+ < ξi|· > vi =∑j
cji < ξi|· > vj − cij < ξj , · > vi =∑j
cjiPij (A)− cijP
ji (A)
Then
dP iA =∑j 6=i
Ajiλi − λj
P ij (A) +Aij
λi − λjP ji (A).
Finally we note that A = A.X(α) giving
Aij =< ξi|A.X(α)vj >=< ξi|λiX(α)vj >= λiXij(α).
Giving
dP i(X)(α) =∑j 6=i
λjλi − λj
Xji (α)P ij (A) +
λiλi − λj
Xij(α)P ji (A).
�
In [4], Goldman observed that although the Hamiltonian Hfα of an invariant function fα was diffi-cult to calculate, one could calculate its Poincare dual Hfα∩ [π] and use the geometric intersectionformula (for homology with coefficients in a flat bundle) to calculate cup products. We will usethe same approach to describe H{liα, l
jβ}∩ [π], the Poincare dual of the Hamiltonian of the Poisson
bracket. In order to do this, we need to review some homology theory.
13
6.2 Homology with coefficients in a flat bundle
For classical homology theory on manifolds, we have given an closed oriented n-manifold M , wehave the fundamental class [π] ∈ Hn(M) and the Poincare dual isomorphism P = ∩[M ] : Hk(M)→Hn−k(M). If S, T are oreinted submanifolds of M with dim(S) + dim(T ) = n intersecting trans-versely, then we let S · T be the algebraic intersection number, i.e. the number of points ofintersection with orientation. Then the algebraic intersection pairing is given by extending this tohomology classes by linearity to obtain Hk(M) × Hn−k(M)
See Dold [3] for details of classical algebraic topology. In particular if [φ] ∈ Hk(M), [ψ] ∈ Hn−k(M)have Poincare duals represented by submanifolds S, T which intersect transversely then
([φ] ∪ [ψ]) ∩ [π] =∑
p∈S∩Tεp(S, T )
where εp(S, T ) = ±1 depending on the orientation of given by Tp(S)⊕ Tp(T ) = Tp(M) at p.
We now consider homology with coefficients in a flat bundle (see Steenrod [13, Chapter 31] fordetails). In particular, the above intersection formula generalizes to homology with coefficients ina flat bundle in a natural way (see Cohen [2]).
Given a vector bundle E → M over an n-manfold M , then the bundle is flat if there is an opencovering for M such that the transition maps for the bundle are constant on each open set in thecovering. Then a section s : M → E is flat if its graph is locally constant. We let Cn(M) are thesingular n-chains for M , given by smooth maps σ : ∆k → M where ∆k is the standard k-simplex.Then the complex of singular chains with coefficients in E is denoted by Cn(M,E) and is the setgenerated by maps σ : ∆k → M with a flat section s over σ(∆k). We denote this by σ ⊗ s anddefine the boundary operator
∂(σ ⊕ s) =k∑i=0
(−1)i∂iσ ⊕ si
where ∂iσ is the σ restricted to the i-th face, and si is the restriction of s to that face. Thepair (C∗, ∂) is easily seen to be a Chain complex and the associated homology groups are denotedHk(M,E). The cohomology of M with coefficients in E can be similarly defined (see [13]).
To define the cup and cap product for coefficients in a flat bundle we need to have a pairingfor the coefficients. Specifically if we have flat bundles E1, E2, E3 over M and a bilinear pairingB : E1 × E2 → E3 then we have
∪B : Ci(M,E1)× Cj(M,E2)→ H i+j(M,E3) ∩B : Ci(M,E1)×Hj(M,E2)→ Hj−i(M,E3).
These maps naturally descend to cohomology and homology.
∪B : H i(M,E1)×Hj(M,E2)→ H i+j(M,E3) ∩B : H i(M,E1)×Hj(M,E2)→ Hj−i(M,E3).
We recall that the algebraic intersection number. If A,B are k, (n − k) submanifolds of M whichintersect transversely, then we have A =
The homology of groups with twisted coefficients is closely related to homology with coefficientsin a flat bundle, Specifically, if π is a group and ρ : π → V is a representation then we define thecochains Ck(π, Vρ) to be generated by maps φ : πk → V . The boundary operator is then definedby
The associated cohomology is denoted H∗(π, Vρ) and similarly for the homology groups H∗(π, Vρ).
Given a representation ρ : π1(M) → V we can define a flat bundle M ×ρ V = M × V/π1(M)over M , where M is the universal cover of M and π1(M) acts on M × V by g(x, v) = (gx, ρ(g)v).Similarly given a flat bundle E over M then there is a representation ρ : π1(M)→ V such that E isisomorphic to the associated flat vector bundle M×ρV →M . Therefore we denote the cohomologyand homology by H∗(M,Vρ), H
∗(M,Vρ). If M is contractible, then there is a natural isomorphismbetween H∗(M,Vρ), H
∗(M,Vρ) and H∗(π1(M), Vρ), H∗(π1(M), Vρ).
6.3 Calculating H{liα, ljβ} ∩ [π], the Poincare dual of the Hamiltonian
We now describe Goldman’s approach to calculating the Poincare dual of the Hamiltonian of aninvariant function ( see [4]). We fix G = PSL(n,R), g = sln and π = π1(S). We further consider πacting on g via Ad ◦ ρ where ρ : π1(S)→ G is a fixed representation.
As S is contractible, from the above, we have T[ρ]R(S,G) ' H1(π, g) ' H1(S, g).
We let B : g→ g∗ be the pairing coming from B. Then for ξ ∈ H1(π, g), B ◦ ξ ∈ H1(π, g∗) and wedefine a map tB : H1(π, g∗)∗ → H1(π, g)∗ by tB(ν)(ξ) = ν(B ◦ ξ).
We also consider the pairing H1(π, g∗)×H1(π, g)→ R given by the cap product and evaluation byB.
H1(π, g∗)×H1(π, g)∩−→ H0(π,R) = R
This gives a map η : H1(π, g)→ H1(π, g∗)∗ with η(Y )(X) = X ∩ Y .
We let θ : H1(π, g)→ H1(π, g∗)∗ be the isomorphism given by the pairing
H1(π, g)×H1(π, g∗)∪−→ H2(π,R)
∩[π]−−→ H0(π,R) = R.
15
Thus for X ∈ H1(π, g∗) thenθ(Y )(X) = (X ∪ Y ) ∩ [π].
Lemma 4 (Goldman, [4]) The diagram below is commutative.
H1(π, g)∩[π]
> H1(π, g)
H1(π, g)∗
ω∨
<tB
θ
>H1(π, g∗)∗
η∨
Given A,B ∈ Hyp recall that
M ij(A,B) =∑k 6=i
(λk(A)
λi(A)− λk(A)P i(A)P j(B)P k(A) +
λi(A)
λi(A)− λk(A)P k(A)P j(B)P i(A)
)
Before we prove calculate H{liα, ljβ} ∩ [π], we have the following lemma.
We define the 1-chain with local coefficients Cij ∈ C1(S, gAdρ) by
Cij =∑p∈α∩β
εp(α, β)(α⊗M ij
ρp(αp, βp) + β ⊗M jiρp(αp, βp)
).
17
As d(α⊗ V ) = α(0)⊗ (Ad(ρ(α))V − V ), then
dCij =∑p∈α∩β
εp(α, β)p⊗(
(Ad(ρp(αp))Mijρp(αp, βp)−M
ij(αp, βp)) + (Ad(ρp(βp))Mjiρp(αp, βp)−M
jiρp(αp, βp))
).
Then by lemma 5, dCij = 0 and Cij defines an element in H1(S, gAdρ). It follows from the definitionof tB, η that
d{liα, ljβ} = tB ◦ η
∑p∈α∩β
εp(α, β)(α⊗M ij
ρp(αp, βp) + β ⊗M jiρp(βp, αp)
) .
Therefore by the commutative diagram in lemma 4 we have
H{liα, ljβ} ∩ [π] =
∑p∈α∩β
εp(α, β)(α⊗M ij
ρp(αp, βp) + β ⊗M ji(βp, αp)).
�
6.4 Calculation of Triple bracket {{liα, ljβ}, lkγ}
We now want to use the above to find a formula for the triple bracket of length functions. We firstdefine the following. Let A,B,C,D ∈ Hyp ⊆ SL(n,R). Then we recall that
bijkl(A,B,C,D) = Tr(P i(A)P j(B)P k(C)P l(D)).
We now prove the generalized sine formula of Theorem 7.
Proof of Theorem 7:
We have{{liα, l
jβ}, l
kγ} = ω(H{liα, l
jβ}, Hl
kγ) = B(H{liα, l
jβ} ∪Hl
kγ) ∩ [π] =
B((H{liα, ljβ} ∩ [π]) · (Hlkγ ∩ [π])).
We haveHlkγ ∩ [π] = γ ⊗ (Lk(ρp(γp)) = γ ⊗W k
where W k is a flat section over γ given by Wp = (Lk(ρp(γp)).
We let p ∈ α ∩ β. Then α ⊗M ijρp(αp, βp) ∈ C1(S, g) denotes the chain α ⊗ U ij(p) where U ij(p) is
a flat section over α with U ij(p)p = M ijρp(αp, βp). Similarly we define V ji(p) the flat section over β
with V ji(p)p = M jiρp(αp, βp). Also by the above
H{liα, ljβ} ∩ [π] =
∑p∈α∩β
εp(α, β)(α⊗ U ij(p) + β ⊗ V ij(p)
).
From the formula for intersection for homology with coefficients in a flat bundle, we then have
{{liα, ljβ}, l
kγ} =
∑p∈α∩β
εp(α, β)B(
(γ ⊗W k).(α⊗ U ij(p) + β ⊗ V ji(p)
))
18
=∑p∈α∩β
εp(α, β)
∑q∈α∩γ
εq(α, γ)B(U ij(p)q,Wkq ) +
∑r∈β∩γ
εr(β, γ)B(V ji(p)r,Wkr )
.
We note that
U ij(p)p =∑l 6=i
λl(α)
λi(α)− λl(α)P i(ρp(αp))P
j(ρp(βp))Pl(ρp(αp))+
λi(α)
λi(α)− λl(α)P l(ρp(αp))P
j(ρp(βp))Pi(ρp(αp)).
We parallel translate along α from p to q. We let αpq, βpq be the curves based at q obtained byconjugating by the arc αpq from p to q along α. Then
Therefore
U ij(p)q =∑l 6=i
λl(α)
λi(α)− λl(α)P i(ρq(αpq))P
j(ρq(βpq))Pk(ρq(αpq))+
λi(α)
λi(α)− λl(α)P k(ρq(αpq))P
j(ρq(βpq))Pi(ρq(αpq)).
As [αpq] = [αq] ∈ π1(S, q) we have
U ij(p)q =∑l 6=i
λl(α)
λi(α)− λl(α)P i(ρq(αq))P
j(ρq(βpq))Pk(ρq(αq))+
λi(α)
λi(α)− λl(α)P k(ρq(αq))P
j(ρq(βpq))Pi(ρq(αpq)).
Similarly for V ji(p)r, we let βpr, αpr be the curves based at r obtained by conjugating by the arc
from p to r along β and note that [βpr] = [βr] ∈ π1(S, r).
Now calculating the sections we get{{liα, l
jβ}, l
kγ} =∑
l 6=i
∑(p,q)∈α∩β×α∩γ
εp(α, β)εq(α, γ)
(λl(α)
λi(α)− λl(α)B(P i(ρq(αq))P
j(ρq(βpq))Pk(ρq(αq)), L
k(ρq(γq)))+
λi(α)
λi(α)− λl(α)B(P k(ρq(αq))P
j(ρq(βpq))Pi(ρq(αq)), L
k(ρq(γq))
))+∑
l 6=j
∑(p,r)∈α∩β×β∩γ
εp(α, β)εr(β, γ)
(λl(β)
λj(β)− λl(β)B(P j(ρr(βr))P
i(ρr(αpr))Pl(ρr(βr)), L
k(ρr(γr)))+
λj(β)
λj(β)− λl(β)B(P l(ρr(βr))P
i(ρr(αpr))Pj(ρr(βr)), L
k(ρr(γr))
))Also as Lk(A) = P k(A) − 1
nI, we have B(X,Lk(A)) = Tr(XLk(A)) = Tr(XP k(A)). Thereforehave
{{liα, ljβ}, l
kγ} =∑
l 6=i
∑(p,q)∈α∩β×α∩γ
εp(α, β)εq(α, γ)
(λl(α)
λi(α)− λl(α)bijlkρq (αq, βpq, αq, γq) +
λi(α)
λi(α)− λl(α)bljikρq (αq, βpq, αq, γq)
)+
19
∑l 6=j
∑(p,r)∈α∩β×β∩γ
εp(α, β)εr(β, γ)
(λl(β)
λj(β)− λl(β)bjilkρr (βr, αpr, βr, γr) +
λj(β)
λj(β)− λl(β)blijkρr (βr, αpr, βr, γr)
)Substituting εp(α, β) = −εp(β, α) we get the final formula.∑
l 6=i
∑(p,q)∈α∩β×α∩γ
εp(α, β)εq(α, γ)
(λl(α)
λi(α)− λl(α)bijlkρq (αq, βpq, αq, γq) +
λi(α)
λi(α)− λl(α)bljikρq (αq, βpq, αq, γq)
)−∑
l 6=j
∑(p,r)∈β∩α×β∩γ
εp(β, α)εr(β, γ)
(λl(β)
λj(β)− λl(β)bjilkρr (βr, αpr, βr, γr) +
λj(β)
λj(β)− λl(β)blijkρr (βr, αpr, βr, γr)
)�
6.5 The PSL(2,R) case
We now show that the above recovers Wolperts Sine formula for n = 2. We consider 3 hyperbolicisometriesA,B,C in H2 with axes in the upper-half plane model having endpoints (a1, a2), (b1, b2), (c1, c2)respectively. Thus considering the action on R2 we can choose e1(A) = (a1, 1), e2(A) = (a2, 1). Fur-ther we have the dual basis
e1(A)(x, y) =x− a2ya1 − a2
, e2(A)(x, y) =x− a1ya2 − a1
.
Similarly for B,C. Then we consider calculating {{lα, lβ}, lγ} In our calculation we therefore onlyneed to calculate
)We now let A be given by the oriented endpoints (a1, a2) = (0,∞). Then
Tr(P1(A)P1(B)P2(A)P1(C)) =−c1c2
(b1 − b2)(c1 − c2)
We consider an oriented geodesic G in the upper-half plane model for H2 with endpoints (u, v).Then G intersects the geodesic A at angle θp (measured counterclockwise from A to G) and heighth where
sin θp =2√−uv
|u− v|h =√−uv
20
We have if p = A∩G we define ε(A,G) to be the orientation of A and G the the intersection point.Then we have ε(A,G) = sgn(u− v), giving
2√−uv
u− v= ε(A,G) sin θp.
Therefore if the axes of B,C intersect the axes of A in angles θB, θC and at heights hB, hC respec-tively then
Tr(P1(A)P1(B)P2(A)P1(C)) =−c1c2
(b1 − b2)(c1 − c2)=
1
4
(√−c1c2√−b1b2
)(2√−b1b2
b1 − b2
)(2√−c1c2
c1 − c2
).
Thus
Tr(P1(A)P1(B)P2(A)P1(C)) =1
4
hChB
ε(A,B)ε(A,C) sin θB sin θC
where p = A ∩B, q = A ∩ C. Therefore
Tr(P1(A)P1(B)P2(A)P1(C)) =1
4ε(A,B)ε(A,C)elBC sin θB sin θC (2)
where lBC is the hyperbolic distance along the axis of A between the points p, q. Similarly
Tr(P2(A)P1(B)P1(A)P1(C)) =1
4
hBhC
ε(A,B)ε(A,C) sin θB sin θC =1
4ε(A,B)ε(A,C)e−lBC sin θB sin θC
Figure 2: Curves αq, βpq, γq with both εp(α, β), εq(α, γ) = +1
We note that if A ∈ SL(2,R) is hyperbolic with hyperbolic translation lA then it λ1(A) =elA/2, λ2(A) = e−lA/2. For α, β, γ homotopy classes of closed curves in S, we can choose theirgeodesic representatives which we also label the same. We have also that (see figure 2)
Therefore plugging this into the above formula we get
{{l1α, l1β}, l1γ} =
1
4
∑p∈α∩βq∈α∩γ
(εp(α, β)εq(α, γ))2
(e−lα/2
elα/2 − e−lα/2elpq sin θp sin θq +
elα/2
elα/2 − e−lα/2e−lpq sin θp sin θq
)−
∑p∈β∩αr∈β∩γ
(εp(β, α)εr(β, γ))2
(e−lβ/2
elβ/2 − e−lβ/2elpr sin θp sin θr +
elβ/2
elβ/2 − e−lβ/2e−lpr sin θp sin θr
)
=1
4
∑p∈α∩βq∈α∩γ
(elpq + elα−lpq
elα − 1
)sin θp sin θq −
∑p∈β∩αr∈β∩γ
(elpr + elβ−lpr
elβ − 1
)sin θp sin θr
=1
4
∑p∈α∩βq∈α∩γ
(elpq + elqp
elα − 1
)sin θp sin θq −
∑p∈β∩αr∈β∩γ
(elpr + elrp
elβ − 1
)sin θp sin θr
We note that {f, g}wp = 1/2{f, g} and lα = 2l1α giving
{{lα, lβ}wp, lγ}wp = 2{{l1α, l1β}, l1γ} =
=∑p∈α∩βq∈α∩γ
(elpq + elqp
2(elα − 1)
)sin θp sin θq −
∑p∈β∩αr∈β∩γ
(elpr + elrp
2(elβ − 1)
)sin θp sin θr
giving the original Wolpert formula.
7 Convexity of Spectral length functions
In [14], Wolpert used his formula for the second derivative to give a new proof that length functionson Teichmuller space were convex along the twist vector fields. This was originally shown byKerckhoff in his proof of the Nielsen realization conjecture (see [8]). Specifically Wolpert observedthat if lα is the geodesic length function of a curve α on S and tβ is the twist vector field associatedwith a simple curve β, then by Theorem 1
t2βlα = {{lα, lβ}wp, lβ}wp =∑
(p,q)∈α∩β×α∩β
(elpq + elqp
2(elα − 1)
)sin θp sin θq > 0
In the Hitchin case, the functions l1α play an important role in understanding the behavior ofrepresentations ρ. In particular l1α([ρ]) = log Λ(ρ(α)) where Λ(A) is the spectral radius of the lineartransformation A.
22
We now consider the convexity of these spectral length functions along the Hamiltonian flow lines.If tβ is the Hamiltonian vector field of l1β then we have
t2βl1α = {{l1α, l1β}, l1β}.
Before we state our convexity result, we will need the following important theorem of Labourie.
Theorem 10 (Labourie [9]) If ρ ∈ Hn(S), then there exists a unique ρ-equivariant continuous mapξρ : ∂∞π1(S)→ Fn where Fn is the space of complete flags in Rn, such that:
1. (Proximality) If α ∈ π1(S)−{1}, and Vi(ρ(α)) is the eigenline for eigenvalue λi(ρ(α)) of ρ(α)then
ξiρ(α+) =i⊕
j=1
Vj(ρ(γ))
for all i, where α+ ∈ ∂∞π1(S) is the attracting fixed point of α.
2. (Hyperconvexity) If x1, . . . , xk,∈ ∂∞π1(S) are distinct and m1 + . . .+mk = n, then
m⊕i=1
ξmi(xi) = Rn.
We now state and prove our convexity result.
Theorem 11 Let α, β ∈ π1(S) with β simple. Let α, β be freely homotopic to transversely inter-secting curves α, β such that for all p, q ∈ α ∩ β we have εp(α, β)εq(α, β) = +1. Then the lengthfunction l1α is strictly convex along the Hamiltonian vector field tβ = Hl1β. In particular if α, β
intersect exactly once, then l1α is strictly convex along the Hamiltonian flow of l1β.
Proof: We apply the formula for the Poisson triple bracket for {{l1α, l1β}, l1β}. We let α, β be closed
curves representing α, β intersecting transversely. The second summation in {{l1α, l1β}, l1β} is over
(p, r) ∈ β ∩ α× (β ∩ β). As β is simple then β ∩ β = ∅ and we have
t2βl1α =
∑k 6=1
∑(p,q)∈α∩β×α×β
εp(α, β)εq(α, β)
(λk(α)
λ1(α)− λk(α)b11k1ρq (αq, βpq, αq, βq) +
λ1(α)
λ1(α)− λk(α)bk111ρq (αq, βpq, αq, βq)
)To show positivity we consider the terms of the summation. We let
t2βl1α =
∑k 6=1
∑(p,q)∈α∩β×α∩β
C(k, p, q)
and study the positivity of the terms C(k, p, q). We have as λ1(α) > λk(α) for k > 1 then
λk(α)
λ1(α)− λk(α)> 0
λ1(α)
λ1(α)− λk(α)> 0.
23
We now consider the functions b11k1, bk111. We let α = [αq], β = [βpq], γ = [βq] ∈ π1(S, q) andρ : π1(S, q)→ PSL(n,R) be given by ρ = ρq. We let ξ : π1(S, q)→ Fn be the limit curve associatedwith ρ. We have that the points α−, α+ ∈ ∂∞π1(S) split each of the the pairs β−, β+ and γ−, γ+.
We set A = ρ(α), B = ρ(β), C = ρ(γ) and choose bases for A,B,C and corresponding dual bases.Then
If < ek(A)|e1(C) >= 0 then e1(C) is in the hyperplane < e1(A), . . . , ek−1(A), ek+1(A), . . . , ed(A) >.Thus again we have
ξ1(γ+) ⊆ ξk−1(α+)⊕ ξd−k(α−).
andξ1(γ+)⊕ ξk−1(α+)⊕ ξd−k(α−) 6= Rd
This contradicts the hyperconvexity in Theorem 10. It follows also that < e1(A)|e1(B) > and< e1(C)|e1(A) > are also non-zero.
We now show < e1(B)|ek(A) >6= 0. To show that this is non-zero, we will use the contragredientinvolution τ : Hn(S)→ Hn(S). Given a representation, we define the contragedient representationρc by ρc(γ) = ρ(γ−1)t. Then τ is defined by τ([ρ]) = [ρc]). The map τ preserves the Hitchincomponent as τ preserves the Fuchsian locus.
We let ρc be the contragredient representation of ρ above and let ξc be its limit curve. We notethat
ρ(γ−1) = C−1 =∑i
λ(γ)−1P i(C).
It follows thatC = ρc(γ) = (C−1)t =
∑i
λi(γ)−1P i(C)t
Thus λi(C) = λd−i(C)−1 and P i(C) = P d−i(C)t. We let vi(C) be defined by
vi(C).w =< ei(C)|w > .
ThenP i(C)w = (vi(C).w)ei(C) = (ei(C)vi(C)t)w
Thus as a matrix we have for w ∈ Rn then
P i(C)t = vi(C)ei(C)t.
We choose basis ei(C) = vd−i(C) for C. Then we vi(C) = ed−i(C) from above.
We have that the function b11k1(ρ(α), ρ(β), ρ(α), ρ(γ)) defines a continuous non-zero function onHn(S). Therefore it must have constant sign. In order to calculate the sign, we will consider ρ tobe on the fuchsian locus.
If ρ is on the fuchsian locus then by definition, ρ = in ◦ρ0 where ρ0 is fuchsian and in : PSL(2,R)→PSL(n,R) is the irreducible representation. The representation in is defined in terms of the Sym-metric product functor. Specifically if V is a vector space and A : V → V a linear map, then SkVis the vector space of symmetric products of k elements of V . Then the map SkA : SkV → SkVis defined naturally. If V = R2 then SkV ' Rk+1 and in : PSL(2,R) → PSL(n,R) is defined byin([A]) = [Sn−1A] for A ∈ SL(2,R).
For any α ∈ π1(S) we have
ρ0(α) = A0 = λ(A0)P1(A0) + λ−1(A0)P
2(A0).
We further let e1(A0), e2(A0) be a chosen basis with dual bases e1(A0), e2(A0). Then the represen-
tation ρ is the action of ρ0 on the (n − 1) symmetric product Sn−1(R2). Thus if A = ρ(α) thenA has eigenvalue bases ek(A) = e1(A0)
n−kek−12 (A0) and dual basis ek(A) = e1(A0)n−ke2(A0)
where ε(X,Y ) is the orientation of the axes of X,Y at the intersection point a and θa the anglebetween the axis X and Y at a measured counterclockwise. Also by equation 3 we have
Tr(P 1(X)P 1(Y )P 2(X)P 1(Z)) = ε(X,Y )ε(X,Z)elab sin θa sin θb
where the axes of X,Y meet at point a and axes X,Z meet at point b and lab is the length alongthe axis of X between a, b.
Given X,Y ∈ PSL(2,R) whose axes intersect then ε(X,Y ) is given by the ordering of the end-points X+, X−, Y+, Y− ∈ ∂∞π1(S) = S1 with ε(X,Y ) = +1 if reading counterclockwise they areX+, Y+, X−, Y−. Otherwise, as the axes intersect they must be ordered X+, Y−, X−, Y+ in whichcase ε(X,Y ) = −1. Therefore (see figure 2)
ε(A0, B0) = ε(ρ0(αq), ρ0(βpq)) = εp(α, β)
andε(A0, C0) = ε(ρ0(αq), ρ0(βq)) = εq(α, β)
Thus it follows from equation 4 above that b11k1(A,B,A,C) has the same sign as
