http://homepages.warwick.ac.uk/~masbb/Papers/41-wolpformula.pdf

PACIFIC JOURNAL OF MATHEMATICS
Vol. 197, No. 1,2001
AN EXTENSION OF WOLPERT’S DERIVATIVE FORMULA
Caroline Series
A well known formula of Wolpert expresses the derivative
of length of a geodesic γon a hyperbolic surface with respect
to the Fenchel Nielsen twist about a fixed simple geodesic C
as a sum of the cosines of the intersection angles between the
two curves. We derive an analogous formula for the derivative
of the length of γwith respect to variation of the length of
the fixed curve C.
1. Introduction.
Let Σbe an oriented hyperbolic surface and let Cbe a simple closed geodesic
on Σ. The Fenchel Nielsen twist about Cis a deformation of Σin which
the surface is cut along Σand reglued after twisting one boundary curve
distance trelative to the other. Let γbe a geodesic transverse to C, with
hyperbolic length l(γ); measured relative to a suitable base point, l(γ) is
a function of the twist parameter t. Wolpert [10], see also Kerckhoff[5],
proved the following well known formula for the derivative with respect to
t:
dl(γ)/dt =Σk
j=1 cos θj.(1)
Here the sum is over the intersection points Qjof Cand γ, and θjis the
angle, measured anticlockwise, from γto Cat Qj.
The formula has been extended to a 3-dimensional situation by Kourouni-
otis [6]. Starting from a Fuchsian group uniformising the surface Σ, the
Fenchel Nielsen twist along Cextends to a quakebend with the now complex
parameter t∈C, where "tis the twist as above and #tis the bending an-
gle. This produces a family of deformations ρt:π1(Σ)→P SL(2,C). The
geodesic γon Σcorresponds to a conjugacy class [γ] in π1(Σ) and hence in
ρt(π1(Σ)). The length l(γ) extends to the complex translation length, also
denoted l(γ), of any element in the conjugacy class ρt([γ]). The derivative
formula becomes
dl(γ)/dt =Σk
j=1 cosh qj,(2)
where now qjis the complex distance (measured with suitable a convention
on orientations) in H3between a pair of axes ˜
C(t) and ˜γ(t) of elements in
223

224 CAROLINE SERIES
the conjugacy classes of ρt([C]) and ρt([γ]), chosen so that the projections
on Σof the time 0 lifts ˜ C(0) and ˜γ(0) intersect in the point Qj.
It is natural to ask for the analogy of Wolpert’s formula for the derivative
of l(γ) with respect to the hyperbolic length l=l(C) of the geodesic C.
In this article we find such a formula and, more generally, find a similar
expression for the derivative of l(γ) with respect to ζ∈C, for any family
of holomorphic deformations ρζ:π1(Σ)→P SL(2,C). The formula is ex-
pressed in terms of the derivatives with respect to ζof the complex lengths
lCand the complex twist parameters tCof curves Cin a fixed pants decom-
position of Σ. The parameters (lC/2, tC) are the complex Fenchel Nielsen
coordinates for quasifuchsian space, as introduced by Kourouniotis [7] and
Tan [9].
To state the formula, we need a more detailed description of the trajectory
of the geodesic γrelative to a fixed pants decomposition of Σ. For simplicity,
we shall describe the case of variation from a base point which is Fuchsian;
the extension to the quasifuchsian setting is explained in 4.1. It is, however,
convenient to state the formula in terms of the complex distance d(L, M)
between oriented lines in H3. The details of our conventions are explained
in Section 2.1 below.
Suppose then that we are given a fixed decomposition of Σinto pairs of
pants by a family of disjoint geodesics C. The trajectory of the oriented
geodesic γcuts in order a sequence Ci1, . . . , Cik+1 of pants curves Cij∈C
in points Qj, so that each adjacent pair Cij, Cij+1 are boundary curves of a
unique pair of pants Πj, numbered such that Qk+1 =Q1and such that the
arc from Q1to Qk+1 is exactly γ. The idea is that there is a unique curve
homotopic to γwith no backtracking, made up of arcs which wrap around a
pants curve, alternating with arcs which cross the pants from one boundary
to the next following the common perpendiculars between the boundaries.
More precisely, let ˜γbe the lift of γthrough a fixed lift ˜ Q1of Q1. Then ˜γ
intersects successively lifts ˜ Cjof Cijin points ˜
Qjwhich project to Qj; in par-
ticular ˜ Qk+1 is the image of ˜
Q1under the covering translation corresponding
to ˜γ.
Let ˜ Djdenote the geodesic containing the common perpendicular to ˜
Cj
and ˜ Cj+1, oriented from the point Q−
jwhere it meets ˜
Cjto the point Q+
j+1
where it meets ˜ Cj+1. Orient ˜ Cjso that the angle from ˜γto ˜ Cjis anti-
clockwise (relative to a fixed given orientation on Σ). Clearly, γis freely
homotopic to the projection on Σof the path made up of geodesic arcs from
Q−
jto Q+
j,Q+ jto Q− j+1,j= 1, . . . , k.
Define dj=d(˜
Cj,˜ Cj+1) and qj=d(˜γ,˜ Cj).Define sj=±d(˜ Dj,˜ Dj+1),
where we choose the + sign if Q−
jis to the left of Q+
jon ˜
Cj(relative to the
orientation induced by the orientation of Σ), and the minus sign otherwise.
Likewise, define pj=±d(˜γ,˜
Dj), where we choose the + sign if ˜γintersects

WOLPERT’S DERIVATIVE FORMULA 225
the geodesic extension of ˜ Djor if it does not intersect and the common
perpendicular from ˜γto ˜ Djis oriented in the same sense as ˜
Cj, and the −
sign otherwise. (In fact dj, sj, pj, qjare really signed complex distances, that
is, distances whose real parts are counted positive or negative depending on
position relative to the orientation of the line along which they are measured,
as explained in the next section. The signs of pj, qjdo not matter in the
formula 3below.)
The function djdepends only on the geometry of Πj, see Section 3.1, and
is holomorphic in the half lengths lC/2. The shift function sjis of the form
njlij/2 + ej+tijwhere nj∈Z,lijis the length l(Cij) of Cij,tijis the asso-
ciated Fenchel Nielsen twist and ejsatisfies 0 ≤ "ej<"lij/2. The function
ejdepends only on the geometry of Πjand Πj−1and is again holomorphic
in the parameters lC/2. The choices of nj, dj, ejare independent of the hy-
perbolic structure on Σ, and depend only on the topology of γrelative to
the pants decomposition C. Details are explained in Section 4.2.
Letting "denote derivative with respect to ζ, our derivative formula is:
l!(γ) = Σk
j=1cosh pjd" j+Σk j=1cosh qjs" j.(3)
In the special case in which the lengths l(Ci), and hence the functions dj
and ej, are real valued and constant, we recover Wolpert’s formula (1).
We give two proofs of Equation (3). The first is very similar to Kerckhoff’s
proof of (1) in [5] and the second is inspired by the proof of (1) in [2],
in which ρt(γ)∈SL(2,C) is expressed in terms of a product of twists
associated to the distance tCquakebend along C. In the present case, we
write ρζ(γ) as a product of translations along the axes of the pants curves ˜
Cj
by the shift distances sj, and translations along the common perpendiculars
˜
Djby distances dj. None of these translations are elements in ρζ(π1(Σ));
nevertheless once ρζ(γ) is expressed in this way, our formula drops out.
The plan of the paper is as follows. In Section 2we discuss signed complex
distance and review the trigonometry of right-angled hyperbolic hexagons,
in particular obtaining a formula for the derivatives of the side lengths. In
Section 3we review hyperbolic pairs of pants and Fenchel Nielsen coordi-
nates in both the two and three dimensional settings. In Section 4we discuss
how to keep track of geodesic paths and give careful definitions of the terms
entering into (3). Finally in Section 5we prove Equation (3) following the
Kerckhoffmethod and in Section 6give the version inspired by [2].
2. Hyperbolic trigonometry.
In this section, following [3], we express trigonometric formulae in terms of
the signed complex distance between oriented geodesics in hyperbolic 3-space
H3. Since the detailed conventions are important in our formulae, we give
all the definitions here.

226 CAROLINE SERIES
2.1. Complex distance. The signed complex distance between oriented
geodesics in hyperbolic 3-space H3, relative to an oriented common perpen-
dicular, should be thought of as the analogue of signed distance measured
along an oriented axis in H2. It is defined as follows, see [3, V.3.].
Let αbe any oriented line in H3, and let P1, P2∈α. Let d(P1, P2)≥0
denote the hyperbolic distance between P1and P2. We define the signed
real hyperbolic distance δα(P1, P2) as d(P1, P2) if the orientation of the arc
from P1to P2coincides with that of αand −d(P1, P2) otherwise.
Now let L1, L2⊂H3be oriented lines with distinct endpoints on the
Riemann sphere ˆ C, with oriented common perpendicular αmeeting L1, L2
in points Q1, Q2respectively, where if L1, L2intersect we take Q1=Q2. Let
vibe tangent vectors to the positive directions of Liat Qi, i = 1,2. Let Π
be the hyperbolic plane through Q2orthogonal to αand let w1denote the
parallel translate of v1along αto Q2. Let nbe a unit vector at Q2pointing
in the positive direction α. The signed complex distance between L1and L2
is
δn(L1, L2) = δα(L1, L2) = δα(Q1, Q2) + iθ
where θ, measured modulo 2πZ, is the angle between w1and v2measured
anticlockwise in the plane spanned by w1,v2and oriented by n. Normalising
so that αis the imaginary axis oriented from 0 to ∞and L1, L2are the
oriented geodesics from −1 to 1, and −reiθto reiθrespectively (r > 0), we
have δα(L1, L2) = log r+iθ. Notice that the choice ±θis independent of
whether or not r > 1. In terms of cross ratio,
exp δα(L1, L2) = (v2−a−)(v1−a+)
(v2−a+)(v1−a−),
where v1, v2are the positive endpoints of L1, L2and a−, a+are respectively
the negative and positive endpoints of αon ˆ C.
For any line oriented L⊂H3, denote by L−1the line Lwith opposite ori-
entation. We have δα(L1, L2) = −δα(L2, L1), δα−1(L1, L2) = −δα(L1, L2)
and δα(L−1
1, L2) = δα(L1, L−1 2) = δα(L1, L2) + iπ.
Finally we define the (unsigned) complex distance d(L1, L2) as d(L1, L2)
=δα(L1, L2) if "δα(L1, L2)>0, −δα(L1, L2) if "δα(L1, L2)<0 and
i|δα(L1, L2)|otherwise. The relations for unsigned distance become d(L1,L2
)
=d(L2, L1), and d(L−1
1, L2) = d(L1, L−1 2) = d(L1, L2) + iπ. When L1and
L2have a common endpoint Q, we define d(L1, L2) = 0 if the orientations
of L1and L2at Qare the same and d(L1, L2) = iπotherwise.
Suppose that Land Mlie in a common plane Π, oriented by a normal
vector n. If Lintersects Mat an angle θmeasured anticlockwise from L
to M, then δn(L, M) = iθ, while if Land Mare disjoint then δm(L, M) =
p+(iπ, where pis the unsigned perpendicular distance between Land M,
mis a vector oriented from Lto Mand (= 0 if Land Mhave the same
orientation relative to their common perpendicular and (= 1 otherwise.

WOLPERT’S DERIVATIVE FORMULA 227
Finally, suppose that M∈P SL(2,C) is non-parabolic with axis Ax M,
oriented pointing from the repelling to the attracting fixed point. We define
the complex length l(M) as δAx M(L, M(L)), where Lis any oriented line
perpendicular to Ax M. Notice that with this definition, "l(M)>0 so that
in fact l(M) = d(L, M(L)). If Mis parabolic, we define l(M) = 0.
2.1.1. Signed versus unsigned complex distance. With the above def-
initions, the signed complex distance between L1and L2depends on the
orientation of their common perpendicular, while the (unsigned) complex
distance does not. Both signed and unsigned distance take values in C/2πiZ.
Notice that one cannot express the signed complex distance in terms of the
endpoints of L1, L2alone, since any possible formula will fail to see the ori-
entation of the common perpendicular α. In fact, as can easily be checked
by normalising as above,
cosh δα(L1, L2) = cosh d(L1, L2) = 1 + χ
1−χ
where χ=(u1−u2)(v1−v2)
(u1−v2)(v1−u2)and ui, viare respectively the repelling and at-
tracting fixed points of Li. In this formula, the distinction between signed
and unsigned complex distance is lost.
The formulae of hyperbolic trigonometry are often given in terms of un-
signed complex distance, see for example in [8] or [7]. However, it will be
important for us to use signed complex distance for the following reason. If
L1and L2vary so as to move through an intersection point changing their
positions relative to the orientation of their common perpendicular α, then
there is a discontinuity in d(L1, L2). Since we want to vary the positions
of all our lines and to obtain a formula that is holomorphic in the relevant
parameters, we have to use not the complex distance d(L1, L2), but the holo-
morphic variable δα(L1, L2) associated to some pre-specified orientation of
α. See also [4] for a similar use.
2.2. Right-angled hexagons. An oriented skew right-angled hyperbolic
hexagon is a cyclically ordered set of six oriented geodesics Lnin H3indexed
by n∈Z(mod 6), such that Lnintersects Ln+1 orthogonally. (We do
not require all the lines Lnto be distinct, nor do we require them to be
oriented consistently around the hexagon.) Define σn=δˆ
Ln(ˆ
Ln−1,ˆ Ln+1).
The hyperbolic sine and cosine rules, [3] VI.2, are:
Sine Rule.
sinh σ1
sinh σ4
=sinh σ3
sinh σ6
=sinh σ5
sinh σ2
.(4)
Cosine Rule.
cosh σn=cosh σn+3 −cosh σn+1 cosh σn−1
sinh σn+1 sinh σn−1
.(5)