# Probing parton distributions in ep/eA and ultra-peripheral collisions

Spencer R. Klein

###### Abstract

Real or virtual photons are excellent probes of nuclear structure, with a
strong sensitivity to gluon distributions. Photonic reactions can be studied
using ultra-peripheral collisions or at an electron-ion collider. Final states
like dijets or open charm production are directly sensitive to the gluon
distributions in nuclei. Exclusive reactions, like exclusive vector meson
production or deeply virtual Compton scattering (DVCS) go further, requiring
at least two gluons. In the Good-Walker paradigm, coherent exclusive
photoproduction is sensitive to the average nuclear configuration (including
gluonic hot spots), and the Fourier transform of the differential cross-
section $d\sigma/dt$ gives the transverse distribution of partonic targets in
a nucleus. The incoherent photoproduction cross-section is sensitive to
partonic fluctuations, including gluonic hot spots. Some reactions, such as
dijet production, involve multiple momentum scales, and thus may be able to
probe the Wigner distribution of nuclear targets. Finally, incoherent
photoproduction is sensitive to partonic fluctuations; an analysis of $J/\psi$
photoproduction on proton targets found that the data clearly preferred a
fluctuating lumpy proton.

##  1 Introduction

Photoproduction has long been used to study nuclear structure [1, 2]. High-
energy photons are sensitive to the partonic structure of matter at low
Bjorken$-x$ [3]. Although most early photoproduction experiments ran at fixed-
target accelerators, interest has shifted toward higher energies which require
colliders, either ultra-peripheral collisions (UPCs) at hadron colliders [4,
5, 6, 7], or $ep$/$e$A collisions, at HERA or the future U. S. electron-ion
collider [8].

UPCs are the energy frontier for photon physics. Heavy nuclei carry strong
electromagnetic fields which, per Weizsäcker-Williams approach can be treated
as a flux of nearly-real photons. Because the photon flux scales with the
square of the nuclear charge ($Z$), UPCs are most prominent in collisions
involving heavy ions. The maximum photon energy is set by the size of the
nucleus (with radius $R_{A}$) and the ion’s Lorentz boost, $\gamma$. In the
lab frame, the maximum photon energy is $\gamma\hbar c/R_{A}$. For lead nuclei
at LHC energies, this is about 100 GeV, equivalent to 500 TeV in the rest
frame of the target nucleus, giving photon-nucleus center of mass energies up
to 700 GeV per nucleon. For photons emitted by protons, the energies are about
5 times higher, due to the larger $\gamma$ and smaller $R_{A}$. These energies
allow the LHC to probe partons down to Bjorken$-x\approx 10^{-6}$ at moderate
$Q^{2}$.

With heavy nuclei, $Z\alpha$ is large, so it is possible, even frequent, for
multiple photons to be exchanged. This is useful for selecting different
impact-parameter ($b$) distributions, and, with that, different photon energy
spectra. However, it also complicates the selection of ’exclusive’
interactions. Another complication arises because the photon direction is
ambiguous: either nucleus can emit a photon [9]. For exclusive production of a
vector meson with mass $M_{V}$ at rapidity $y$, the two photon energy ($k$)
solutions and associated Bjorken$-x$ values are

$k=\frac{M_{V}}{2}\exp{(\pm y)},\hskip
18.06749ptxm_{p}=\frac{M_{V}}{2}\exp{(\mp y)}$ (1)

where $m_{p}$ is the proton mass. The two-fold ambiguity can be resolved by
analyzing multiple considering classes of events, containing different
requirements on additional photon exchange (ı. e. with different impact
parameter distributions and photon spectra), creating a set of linear
equations that can be solved to find the cross-sections for the two photon
directions. For photoproduction, the $Q^{2}$ scale is set by the invariant
mass of the final state. For charmonium and bottomonium, this hard scale
allows for comparisons with perturbative QCD.

Electron-ion collisions offer several advantages over UPCs. Photons can have a
wide range of virtualty. By tagging the outgoing electron, the photon
4-momentum can be determined, irrespective of the final hadronic state. High-
energy electron-proton collisions were first studied at the HERA $ep$
collider. These studies will be continued by the U. S. electron-ion collider
(EIC). The EIC will accelerate nuclei from protons to uranium, at a very high
luminosity. Protons and light ion beams can be polarized, as can the electron
beam. The detector(s) will be optimized for electron-ion collisions, with
nearly $4\pi$ angular coverage. However, the maximum photon energy is lower
than is available with UPCs at the LHC.

##  2 Experimental Probes

Photons can produce a wide variety of final states. Many of these reactions
can be described using a dipole model, in which a photon fluctuates to a
quark-antiquark dipole, which then interacts hadronically with the target.

One of the simplest reactions, photon-gluon fusion, can lead to dijets or to
open charm. Since only a single gluon is involved, the final state cannot be
color neutral; additional particles must be created by the color string
between the dipole and the hard reaction products. These processes are
theoretically simpler than exclusive reactions, but are more complex
experimentally. The dijet or charm-quark pair masses fill the role of the
vector meson mass in Eq. 1. For dijets, the requirement that the jets be
energetic enough to be well reconstructed limits the reach in $x$. The ATLAS
collaboration has studied UPC photoproduction of jets in lead-lead collisions
[10]. With the kinematic cuts, the analysis was sensitive to the range
$10^{-2}<x<1$ and 1600 GeV${}^{2}<Q^{2}<40,000$ GeV2. The data were consistent
with the nCTEQ nuclear parton distribution functions. Similar analyses are
possible for open charm, offering sensitivity to somewhat lower $x$ and
$Q^{2}$ ranges. Open charm analyses might be able to probe gluons with
Bjorken$-x$ down to $\approx 2\times 10^{-6}$, at $Q^{2}\approx{\rm few}\
m_{c}^{2}$.

##  3 Exclusive vector meson production

In exclusive photoproduction a photon fluctuates to a $\overline{q}q$ dipole,
which then scatters elastically but hadronically from a target nucleus, often
emerging as a vector meson. This elastic scattering can be described as
occurring via Pomeron exchange. In lowest order pQCD, the Pomeron consists of
two gluons. Because this involves elastic scattering, the Pomeron has the same
quantum numbers as the vacuum; the vector meson retains the quantum numbers of
the photon, including helicity. Due to the two gluon exchange, this reaction
can be described in terms of a Generalized Parton Distribution (GPD). However,
it can be related to single-gluon distribution using a Shuvayev transformation
[15]. Generally, one gluon carries most of the momentum, and the other is much
softer.

Recent next-to-leading order QCD calculations of $J/\psi$ production have
found a rather different picture than the LO approach [16]. The quark
contribution is significant, partly because the amplitudes from NLO gluon
diagrams have a different sign than the LO contributors. These data can still
be used in NLO parton distribution fits, but the fits must consider both the
quark and gluon contributions. Also, there is an unexpectedly large scale
uncertainty. For now, this can be controlled by comparing the cross-sections
on proton and heavy-ion targets, to find the nuclear suppression. These issues
also remain in fits to find GPDs using this data.

Measurements of $J/\psi$ photoproduction on proton targets have been made at
the fixed target experiments, HERA, and the LHC, with the latter using $p{\rm
Pb}$ and $pp$ collisions. In $pA$ collisions, the photon is predominantly
emitted by the ion, while for $pp$ collisions, bootstrapping (using other
measurements at lower energy) has been used to solve the two-fold ambiguity
(Eq. 1). The cross-section largely follows a power law, with

$\sigma(W_{\gamma p})\propto W_{\gamma p}^{0.70\pm 0.04}$ (2)

as is shown in Fig. 1 (left) [17] . Except near threshold, the cross-section
is close to a power law up to $W_{\gamma p}=2$ TeV, corresponding to Bjorken
$x$ around $2\times 10^{-6}$ (at $Q^{2}\approx 2.25$ GeV2). At lowest order,
this is expected if the gluon distributions follow a power law. However, at
NLO, the picture must be more complex.

For lead targets, the cross section $J/\psi$ photoproduction is smaller than
for 208 independent nucleons (the impulse approximation prediction)[18, 19,
20] . The suppression factor ($S_{\rm Pb}$) relative to the impulse
approximation quantifies the nuclear shadowing, due to changes in the parton
distributions. As Fig. 1 (right) shows, $S_{\rm Pb}$ is significantly below 1.
It is also below the predictions of the STARlight Monte Carlo, which uses a
Glauber calculation to account for the fact that the nucleons are clumped
together into a single nucleus, so photons hitting the center of the nucleus
may interact with multiple nucleons, indicating that the modifications go
beyond geometric effects. The data are in agreement with dipole model
calculations. There are many different dipole model calculations, using
different models of partons in nuclei, with different gluon shadowing and
saturation, and somewhat different predictions. Unfortunately, the limited
precision of the data and the similarity of different models preclude more
specific conclusions. To make progress, it will be necessary to compare the
theories with multiple observables. The $\psi^{\prime}$ is heaver than the
$J/\psi$, but with a different wave function, so its cross section provides an
additional discriminant. The $\Upsilon$ states should have further
discriminating power. Measurements of $\sigma/dt$ are also sensitive to
saturation, which changes the effective shape of the nucleus [21].

Figure 1: (left) The cross-section for $J/\psi$ photoproduction on proton
targets. From Ref. [17]. (right) $S_{\rm Pb}$, the $J/\psi$ photoproduction
cross-section relative to a proton reference. As the center-of-mass energy
increases, $S_{\rm Pb}$ drops. From Ref. [20].

In the Good-Walker picture, coherent and incoherent production of vector
mesons are respectively sensitive to the average nuclear configuration and
their fluctuations [22, 23]. The coherent cross-section is obtained by summing
the amplitudes for the target to remain in its ground state and then squaring,
while the total cross-section is obtained by squaring all of the amplitudes
and then adding. The incoherent cross-section is the difference - the square
of sums minus the sum of squares, giving a measure of fluctuations. This
approach has been use to study $J/\psi$ photoproduction in $ep$ collisions at
HERA, where a model based on a lumpy proton fit the data much better than a
smooth proton [24]. However, there are two issues with this Good-Walker
approach [25].

First, Good-Walker associates coherent photoproduction only with interactions
where the target remains intact. However, coherent photoproduction has been
observed in two contexts when the target dissociates: photoproduction
accompanied by mutual Coulomb excitation [11, 12], and photoproduction in
peripheral heavy ion collisions [13, 14].

Second, nuclear excitation is endothermic, so requires enough energy transfer
to the target. Lead and gold are very different in this respect. The lowest
excited state in lead is at an energy of 2.6 MeV, while for gold, the lowest
lying state is at 77 keV. So, when the energy transfer is small, we expect
substantial differences in excitation. In contrast, in the Good-Walker based
calculations, lead and gold are similar, with similar nuclear shapes, and
similar parton distributions, so should have similar $d\sigma_{\rm
incoherent}/dt$.

These issues can be at least partially avoided in a semi-classical approach
[25], where one sums the amplitudes for reactions that are indistinguishable
from each other, with a propagator to account for different locations for
different target nucleons. This leads to a similar picture for coherent
photoproduction, but a rather different take on incoherent production. In this
model, a completely static target will still have incoherent interactions,
although a black disk will not.

##  4 GPDs, and the transverse distribution of targets

Although vector meson photoproduction is naturally described using GPDs, the
data does not permit direct extractions, because $\Delta$, the momentum
difference between the two gluons is not known (i. e. we cannot separate
$x_{1}$ and $x_{2}$), so there is not direct access to GPDs. So, recent
studies have focused on a related subject, the spatial dependence of the gluon
distributions. The transverse distribution of targets, $F(b)$ is given by the
Fourier transform of $d\sigma/dt$:

$F(b)\propto\int_{0}^{\infty}p_{T}dp_{T}J_{0}(bp_{T})\sqrt{\frac{d\sigma}{dt}}.$
(3)

Several caveats apply here.

First, the square root converts cross-section to amplitude. So, it is
necessary to flip the sign of $\sqrt{d\sigma/dt}$ when crossing each
diffractive minimum to account for the phase evolution with $t$. This is
theoretically straightforward, but it is not always easy to pinpoint the
minima accurately, especially in the presence of final state radiation.

Second, the $p_{T}$ integral runs from 0 to infinity, but the experimental
data is limited to a finite range. The limited range can introduce windowing
artifacts, since the finite integral gives a Fourier transform that is the
convolution of $d\sigma/dt$ with a box function.

The STAR collaboration used this tomographic approach to determine $F(b)$,
using 394,000 photoproduced $\pi^{+}\pi^{-}$ pairs [26]. Data were collected
using a trigger that required neutrons in both zero degree calorimeters,
signalling breakup of both nuclei, along with two to six hits in a time-of-
flight system that surrounded the STAR time projection chamber. The dipion
mass spectrum was well described by a mixture of $\rho^{0}$, direct
$\pi^{+}\pi^{-}$ and $\omega\rightarrow\pi^{+}\pi^{-}$, from both coherent and
incoherent production. The collaboration measured the incoherent production in
the region $0.2{\rm GeV}^{2}<|t|<0.45{\rm GeV}^{2}$ \- above the $|t|$ range
where coherent production is significant. There, $d\sigma/dt$ was fit to a
dipole form factor (appropriate for a single-nucleon target). An exponential
did not give a good fit to this data. The dipole form factor was extrapolated
to $t=0$ and subtracted from the total $d\sigma/dt$, leaving the coherent
component, shown in Fig. 2 (left). This $d\sigma/dt$ was then Fourier
transformed, following Eq. 3, to give the transverse distribution shown in
Fig. 2 (right). The blue band shows the uncertainty as the maximum $p_{T}$ in
Eq. 3 is varied from 0.05 GeV2 to 0.09 GeV2, showing the effects of windowing.
The maximum $p_{T}$ has a large effect at small $b$, but does not affect the
effective size of the target. The negative wings at $|b|\approx 9$ fermi are
likely due to a contribution from the other nucleus, going in the opposite
direction, which contributes a negative amplitude at large $b$.

There are several issues associated with this first extraction of F(b), beyond
the caveats listed above. The effective nuclear radius is considerably larger
than the radius of gold. This is partly because the measured $d\sigma/dt$
includes contributions from the photon $p_{T}$ and the detector resolution, in
addition to the target. A follow-up study [28] attempted to account for these
factors, but the fits preferred an unphysically large gold radius. The photon
$p_{T}$ spectrum is especially problematic, because it depends on the allowed
range of impact parameters; requiring $XnXn$ mutual Coulomb excitation selects
events with relatively small impact parameters, with $\langle b\rangle\approx
18$ fermi with gold at RHIC [29]. The limited impact parameter range increases
the mean $p_{T}$

The ALICE Collaboration has measured $d\sigma/dt$ for coherent $J/\psi$
photoproduction in lead-lead collisions [30], also finding a $d\sigma/dt$
compatible with a nuclear size larger than the known Woods-Saxon radius, but
consistent with models that include nuclear shadowing or saturation (leading
to a nucleus that is effectively larger). For their study, ALICE removed the
effect of detector resolution and the photon $p_{T}$ (which is much smaller
than in the STAR setup). ALICE has also studied the $p_{T}$ distribution in
incoherent $J/\psi$ photoproduction [31]. Since the presumptive targets are
individual nucleons, the spectrum is harder, but the data indicate that sub-
nucleon fluctuations also play a role.

These studies of the transverse distribution can shed light on GPDs. There is
also another direct GPD measurement, using $J/\psi$ photoproduction on
polarized targets. The left-right asymmetry is sensitive to polarized GPDs.
The STAR Collaboration has already made a first measurement of this asymmetry,
albeit with large errors and a consistency with zero [32].

Looking ahead, theoretical studies have probed how the full Wigner
distribution could be measured in UPCs. Wigner distributions are functions of
two conjugate variables, $p_{T}$ and $b$, so are problematic to observe.
However, we may be able to evade the uncertainty-principle limits, by using
final states that involve two different momentum scales [33]. Photoproduction
of dijets is one example. In the dipole model, the orientation of the dipole
fluctuations follows the photon electric field, which itself follows the
impact parameter vector. The dipole-target cross-section depends on the
spatial gradients of parton density, since for finite dipole separation, the
quark and the antiquark will see slightly different density distributions.
Correlations between the parton $p_{T}$ and transverse position lead to an
azimuthal correlation between the dijet pair $p_{T}$ and the difference
between the two jet $p_{T}$, which is related to the dijet pair mass. This
angular distribution is

$\frac{d\sigma}{d\theta}=v_{0}[1+2v_{2}\cos{2\Delta\theta}]$ (4)

where $\theta$ is the angle between $\vec{p}_{T1}+\vec{p}_{T2}$ and
$\vec{p}_{T1}-\vec{p}_{T2}$.

Figure 2: (left) $d\sigma/dt$ for coherent photoproduction of dipions, for
both $XnXn$ and $1n1n$ nuclear excitations. $1n1n$ corresponds to the case
where each nucleus emits a single neutron, usually via Giant Dipole Resonance
excitation, while $XnXn$ corresponds to observing any non-zero number of
neutrons from each nucleus. The inset shows a blow-up at very small $|t|$,
where destructive interference between the production on the two nuclei [27]
is visible. At slightly larger $|t|$, the data is reasonably well fit to an
exponential, in a limited $|t|$ range. At higher $|t|$, two diffractive minima
are visible. (right) F(b), calculated using Eq. 3 from the data on the left.
The blue band shows the uncertainty as the $|t|$ range for the Fourier
transform is varied. Both figures are from Ref. [26].

##  5 Partonic Fluctuations

As noted above, the Good-Walker approach links $d\sigma_{\rm incoherent}/dt$
to fluctuations in the target, including both variations in the spatial
distribution of nucleons, and also partonic flucutations, such as an
evanescent gluonic hotspots. Although the relationship between $t$ and
distance scales is not as clear as for coherent production, the incoherent
$d\sigma/dt$ is well suited to model testing. Heikki Mantysaari and Bjorn
Schenke fitted coherent and incoherent $J/\psi$ photoproduction data from HERA
to two models [24]. One model was a smooth (Gaussian) proton, while the other
had a fluctuating proton with fluctuating parton configurations. As Fig. 3
shows, the fluctuating proton fit the data, while the smooth proton had a much
smaller incoherent cross section than the data. The figure also shows some
examples of the fluctuating proton.

Figure 3: (left) HERA data on coherent and incoherent production of $J/\psi$,
compared with two models, one with a smooth proton, and the other with a
fluctuating proton. (right) Examples of individual manifestations of the
fluctuating proton. From Ref. [24].

The energy evolution of the incoherent cross-section lends further clues to
nuclear evolution [34]. Low photon energies correspond to higher Bjorken$-x$,
where fluctuations are expected to be less frequent. As the energy rises,
fluctuations should become more common. However, as the energy rises further,
parts of the target will become fully absorptive, and so will no longer be
subject to fluctuations. At high enough energies, the nucleus is expected to
look like a black disk. At this point, the incoherent cross-section should
then be zero. The energy at which the incoherent cross-section is maximal
depends on the mass of the produced vector meson.

This pattern also applies to nuclei. Because of the higher parton density in
heavy nuclei, the maximum in the incoherent cross-section and the threshold
for black disk behavior should be much lower than for proton targets.

##  6 Conclusions

Ultra-peripheral collisions are the energy frontier for photon physics, able
to probe vector meson production with nearly-real photons up to photon-nucleon
center of mass energies of around 2 TeV. Final states like dijets or open
charm are sensitive to the gluon densities in proton or nuclear targets.

Exclusive vector mesons production is slightly more complicated theoretically,
requiring two gluons at lowest order. For proton targets, $\sigma(\gamma
p\rightarrow J/\psi p)$ follows a smooth power-law or near-power law increase,
with no clear turn-over or other structure. $J/\psi$ photoproduction on lead
targets is suppressed compared to a proton target reference, roughly
consistent with the midpoint of nuclear parton distributions, and is
indicative of moderate shadowing.

Because two gluons are exchanged, vector meson photoproduction can be best
described using a GPD. However, the relative energies of the two gluons
($\Delta$) is unknown, so GPDs cannot be directly measured. However, it is
possible to use measurements of $d\sigma/dt$ for coherent photoproduction to
study a related quantity, the transverse distribution of gluons in a nucleus.
$J/\psi$ photoproduction data on lead targets is consistent with a moderate
difference between the measured transverse gluon distribution and the nucleon
distribution in lead, as predicted by models that include shadowing.

Incoherent photoproduction is sensitive to fluctuations in the nuclear
configuration, including fluctuations in parton densities and nucleon
positions. An analysis of coherent and incoherent $J/\psi$ photoproduction
data at HERA found that the cross-sections were consistent with a model where
the proton underwent fluctuations, whereas a smooth proton model greatly
underpredicted the incoherent cross-section.

Looking ahead, the Electron-Ion Collider will provide very high luminosity
$\gamma^{*}p/\gamma^{*}A$ collisions which will be studied in detail with an
optimized detector - ePIC. ePIC will offer nearly $4\pi$ coverage, with an
extensive forward detector designed to observe the products of nuclear
breakup.

I thank Minjung Kim for useful comments on this manusript. This work is
supported in part by the U.S. Department of Energy, Office of Science, Office
of Nuclear Physics, under contract numbers DE-AC02-05CH11231.

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# Rank two free groups and integer points on real cubic surfaces

Greg McShane

###### Abstract.

Counting integer points on the Markoff cubic is closely related to questions
in hyperbolic geometry. In a previous work with Igor Rivin we investigated the
regularity of the geodesic length function for a punctured torus. Here we
extend this work to the three holed sphere.

The author thanks ICERM, Brown University and the CNRS for support whilst
preparing this article.

##  1\. Introduction

The rank two free group $\mathbb{Z}*\mathbb{Z}$ arises as the fundamental
group of exactly four non-homeomorphic surfaces: the three-holed sphere
$\Sigma(0,3)$, the one-holed torus $\Sigma_{1,1}$, the two-holed projective
plane (or one holed mobius band) $C(0,2)$, and the one-holed Klein bottle
$C(1,1)$. Of these the first two are orientable and the second two are
nonorientable. Furthermore $\Sigma_{1,1}$ enjoys the remarkable property that
every automorphism of its fundamental group is geometric ie induced by a
homeomorphism. Equivalently, every homotopy-equivalence
$\Sigma_{1,1}\rightarrow\Sigma_{1,1}$ is homotopic to a homeomorphism. It is
not surprising then that much time and energy has been devoted to the
geometric and topological properties of the one holed torus. Here we will be
corncerned with trying to extend the methods used for the one holed torus to
study the moduli spaces of the other surfaces.

Figure 1. From left to right- the one holed torus, one holed Klein bottle, one
holed Mobius band, three holed sphere.

###  1.1. Hyperbolic surfaces

Each of these four surfaces admits a (non unique) hyperbolic structure, in
fact there is a three dimensional deformation space of such structures. Indeed
it has been known since the time of Fricke that, as a consequence of Rieman’s
Uniformization Theorem, the three holed sphere and the one holed torus the
moduli space of structures can be identified with a semi algebraic subset of
$\mathbb{R}^{3}$. More precisely, one obtains a hyperbolic structure $X$ on
$\Sigma$ from a discrete faithful representation $\rho$ of $\pi_{1}(\Sigma)$
into $\mathrm{isom}(\mathbb{H})$ such that
$\mathbb{H}/\rho(\mathbb{Z}*\mathbb{Z})$ is homeomorphic to $\Sigma$. If
$\Sigma$ is orientable one can lift $\rho$ from a representation into
$\mathrm{isom}(\mathbb{H})^{+}$, which is isomorphic to
$\text{P}\mathrm{SL}(2,\mathbb{Z})$ to a representation,
$\tilde{\rho}:\mathbb{Z}*\mathbb{Z}\rightarrow\text{SL}(2,\mathbb{R})$. Doing
this allows one to consider the trace
$\text{tr\,}\tilde{\rho}(\gamma)\in\mathbb{R}$ for any element $\gamma$ of the
free group so that many problems can be reduced to questions of linear
algebra. In particular one defines a character map

$\chi:X\mapsto(\text{tr\,}\tilde{\rho}(\alpha),\text{tr\,}\tilde{\rho}(\beta),\text{tr\,}\tilde{\rho}(\alpha\beta)),$

which provides an embedding of the moduli space of hyperbolic structures on
$\Sigma$. Goldman [9] studied the dynamical system defined by the action of
the group of outer automorphisms of $\mathbb{Z}*\mathbb{Z}$ on the moduli
space. If $\phi$ is an automorphism then it acts on the representations

$\tilde{\rho}\mapsto\tilde{\rho}\circ\phi^{-1}$

and evidently this induces an action on the points in the image of the
embedding above. It turns out that, by applying Cayley-Hamilton theorem to
$\mathrm{SL}(2,\mathbb{Z})$, it is easy to show that this action is the
restriction of polynomial diffeomorphisms of $\mathbb{R}^{3}$. Later Goldman
and his collaborators [8] made a similar study for the two-holed projective
plane and the one-holed Klein bottle. This latter work is quite delicate for
two reasons. Firstly, the representation $\rho$ no longer lifts to
$\mathrm{SL}(2,\mathbb{R})$ but $\mathrm{SL}(2,\mathbb{C})$ and secondly the
automorphisms will in general no longer be geometric for these surfaces.

###  1.2. Markoff triples

An important aspect of the theory of the action of the group of outer
automorphisms is that it admits an invariant function. In the case of the pair
of orientable surfaces this is

$\kappa:(x,y,z)\mapsto x^{2}+y^{2}+z^{2}-xyz.$

The level set $\kappa^{-1}(0)$ has five connected components - the singleton
$(0,0,0)$ and four codimension one manifolds homeomorphic to a
$\mathbb{R}^{2}$. The non trivial components can be identified with the
Teichmueller space of a once punctured torus. It is well known that there are
integer points in $\kappa^{-1}(0)\cap\mathbb{R}_{+}^{3}$, for example
$(3,3,3)$, and that they form a single orbit for the action of the outer
automorphisms. From such a point one obtains a Markoff triple, that is a
solution in positive integers of the Markoff cubic

$x^{2}+y^{2}+z^{2}-3xyz=0$

simply by dividing through by $3$. This map is natural in that it is a
conjugation of automorphisms of the level set $\kappa^{-1}(0)$ with the
automorphisms of the Markoff cubic. So, since we will not be interested in
arithemetic properties of these solutions, we will also refer in what follows
to elements of $\kappa^{-1}(0)\cap\mathbb{N}^{3}$ as Markoff numbers. An
integer which appears in a Markoff triple is called a Markoff number.

####  1.2.1. Reduction theory

The fact that the Markoff triples form a single orbit is a corollary of a
classical result of Markoff in reduction theory which we now explain briefly.
We start by giving the Markoff triples an ordering, in the obvious way, using
the sup norm on $\mathbb{R}^{3}$ and proceed to show that any solution can be
obtained from $(1,1,1)$ by repeatedly applying generators of the group of
automorphisms of the cubic. This automorphism group can be shown to be
generated by:

  1. (1)

sign change automorphisms $(x,y,z)\mapsto(x,-y,-z)$.

  2. (2)

coordinate permutations eg $(x,y,z)\mapsto(y,x,z)$.

  3. (3)

a Vieta flip $(x,y,z)\mapsto(x,y,3xy-z)$.

Since Markoff triples are solutions in positive integers we will use just the
Markoff morphisms that is the group generated by permutations and the Vieta
flip. Given a Markoff triple $(x,y,z)\neq(1,1,1)$ we may apply a permutation
so that $x\leq y<z$ which one can try to “reduce” using the Vieta flip, that
is replacing the it by the triple $(x,y,3xy-z)$. We obtain a smaller solution
provided $3xy<2z$ and this inequality holds for every Markoff triple except
$(1,1,1)$. The reduction process gives rise to the structure of a rooted
binary tree, the Markoff tree, on the Markoff triples with $(1,1,1)$ as the
root triple.

Figure 2. Level set for the the length function $\ell$.

####  1.2.2. Counting integer points

The question of counting for Markoff triples, that is for $R>0$ estimating the
size of the set of Markoff triples $(x,y,z)$ for which the sup norm is less
than $R$, was first investigated by Gurwood [10] in his thesis. He established
an asymptotic formula using a correspondence between Markoff and Farey trees.
Somewhat later Zagier [15] obtained an improved error term. At the time of
writing the best result is due to a Rivin and the author[11]:

$M(R)=C(\log R)^{2}+O(\log R\log\log R),$

for a numerical constant which has a geometric interpretation.

The approach used in [11] differs from that of Gurwood and Zagier in that it
uses ideas of H. Cohn to interpret the question in terms of geometry of
hyperbolic surfaces. In [3] Cohn shows that the Markoff triple $(3,3,3)$ is
the image under the character map $\chi$ of the modular torus that is
$X=\mathbb{H}/\Gamma$ where $\Gamma$ is the commutator subgroup of
$\text{PSL}(2,\mathbb{Z})$. It is easy to show that the modular surface is a
6-fold cover of the modular surface $\mathbb{H}/\text{PSL}(2,\mathbb{Z})$.
Secondly, he shows that the permutations and the Vieta flips are induced by
automorphisms and concludes that the action of
$\text{Aut}(\mathbb{Z}*\mathbb{Z})$ acts transitively on Markoff triples. Now
identifying $\mathbb{Z}*\mathbb{Z}$ with the fundamental group of the modular
torus one can show that if $x$ is a Markoff number then there exists a simple
loop $\gamma$ such that $x=\text{tr\,}\tilde{\rho}(\gamma)$. It is well known
that $\ell_{X}(\gamma)$, the length of the closed geodesic homotopic to
$\gamma$ on the surface $X=\mathbb{H}/\rho(\mathbb{Z}*\mathbb{Z})$, can be
computed from the trace using the relation

(1) $2\cosh\ell_{X}(\gamma)/2=|\text{tr\,}\tilde{\rho}(\gamma)|,$

so there is an explicit relation between Markoff numbers and lengths of simple
geodesics on the modular torus. In [11] the simple closed curves are embedded
as a subset of primitive elements in the integer lattice
$\mathbb{Z}^{2}\subset\mathbb{R}^{2}$ and the length function
$\gamma\mapsto\ell_{X}(\gamma)$ is shown, using a simple geometric argument,
to extend to a proper convex function
$\ell:\mathbb{R}^{2}\mapsto\mathbb{R}_{+}$. Thus the counting problem for
Markoff numbers can be settled by counting integer points in the set

$\ell^{-1}([0,R])=R\ell^{-1}([0,1])\subset\mathbb{R}^{2}$

It is easy to visualize this set using a very short computer program see for
example Figure 3.

###  1.3. Clebsch cubic

It is natural to study the distribution of integer points on other affine
surfaces $V_{k}(\mathbb{R})$ in the family, for $k\in\mathbb{Z}$

$x^{2}+y^{2}+z^{2}-3xyz=k.$

In one direction Ghosh, Sarnak [4] show that for almost all $k$ the Hasse
Principle holds, namely that $V_{k}(\mathbb{Z})$ is non-empty if
$V_{k}(\mathbb{Z}_{p})$ is non-empty for all primes $p$. However they also
show that there are infinitely many k’s for which it fails. They give a proof
that the Markoff morphisms again act on $V_{k}(\mathbb{Z})$ but with finitely
many orbits rather than a single unique orbit.

There is a surface of particular interest in this family namely

$x^{2}+y^{2}+z^{2}-xyz=20.$

Goldman has shown that this set can be identified with the real points of the
Clebsch cubic but what is of immediate interest to us is that it contains the
integer point $(2,2,-2)$. This point is the image under $\chi$ of a
representation for which the corresponding hyperbolic surface $X$ is a three
punctured sphere. Let us set $x=2$ so that

(2) $16=y^{2}+z^{2}-2yz=(y-z)^{2}$

ans we see that there are infinitely many integer solutions on the lines
$(2,t,t\pm 4),t\in\mathbb{R}$ corresponding to an integer value of the
parameter $t$. An obvious consequence of this observation is that the growth
rate of integer triples is quite different from that of the Markoff triples as
it must be at least linear.

In fact these points fall into one of three orbits under the action of the
Markoff morphisms with the root of the corresponding Markoff tree at
$(2,0,-4),(2,1,-3)$ and $(2,2,-2)$. Using reduction theory Goldman [9] has
shown that, after possibly applying a sign change automorphism, any integer
solution on this surface is in the orbit of one of these three triples under
the Markoff morphisms. Each of the three solutions corresponds to hyperbolic
structures on some surface but only $(2,2-2)$ is non singular. The triple
$(2,1,-3)$ and $(2,0,-4)$ correspond to structures on a punctured disc each
with a single cone singularity of angle $2\pi/3$ and $\pi$ respectively.

If we choose one of the three root triples then, using the same process as was
used to analyse the punctured torus in [11], we can vizualize the level sets
of the corresponding length function.

Figure 3. Level set for the the length function $\ell$ for the three punctured
sphere. The scale is not the same as before as can be seen from the
representation of the black square cross representing the axes.

####  1.3.1. The three punctured sphere.

The first triple we shall consider is $(2,2,-2)$ whose corresponding surface
the three punctured sphere.

One sees in Figure 3 that the set $\ell^{-1}([0,R]$ is quite different from
that obtained in the case of the holed torus. Though it is still symmetric,
that is invariant under half turn round the origin, it is not compact for any
$R>0$. Moreover rather than being convex the function is “piecewise” concave.
The six excursions to infinity correspond to oriented simple loops round one
of the three punctures. For each of these loops there is no minimiser in the
homotopy class for length, informally we say that the geodesic has length
zero, and this is the reason for the infinite spike in this direction.

####  1.3.2. The triple $(0,4,2)$.

The triple $(0,4,2)$ corresponds to a structure on a punctured disc with a
single cone singularity of angle $\pi$. The boundary of the disc is a geodesic
of strictly positive length and, as before, we consider the puncture as being
a geodesic of length $0$. The holonomy round the singularity is elliptic which
means that

  * •

the representation $\tilde{\rho}$ is not injective

  * •

$(0,4,2)$ is fixed by an infinite cyclic subgroup of the Markoff morphisms.

So we see infinitely many spikes as is illustrated in Figure 4.

Figure 4. Level set for the the length function $\ell$

There is similar picture for $(0,3,1)$ which we will not reproduce here.

####  1.3.3. The “generic” case

In general a hyperbolic structure on the three holed sphere will not have
punctures or singularities and the boundary holonomies will be hyperbolic. In
this case the minimum of the absolute value of the three numbers $x,y,z$ in
the triple is strictly greater than $2$. The level set for the length function
resembles a six pointed star with each of the prongs corresponding to a simple
loop round one of the boundary components.

Figure 5. Level sets of the length function $\ell$ for a collection of
equilateral three holed spheres as boundary length increases. The level is
compact, just as for the punctured torus, though it is no longer convex.

In Figure 5 we see the level sets corresponding to several equilateral three
holed spheres. Here by equilateral we mean that the lengths of the three
boundary components are equal. The lengths are actually
$2.02,2.05,2.1,2.3,2.6,3,63$ and 6 and the figure illustrates that as length
increases the star shrinks and seams to be converging to a convex hexagon.

##  2\. Statement of results

This work is concerned with the lengths of curves on oriented surfaces we will
treat the non orientable surfaces elsewhere. The surfaces we treat here then
are hyperbolic pairs of pants which have three boundary curves of positive
length. The techniques apply to some extent to generalized pairs of pants
where the boundary lengths can be zero or purely imaginary complex numbers,
which corresponds to a boundary being replaced by a cone singularity. Our main
result relates the shape of the level sets illustrated in the figures of the
previous paragraph to geometric features of $X$ namely whether or not it has
punctures and conical singularities. What we are most interested in is
understanding why the level set of the length function is (locally) concave as
opposed to being convex for a one holed torus. Very roughly the difference
comes down to the fact that the minimiser for length in a homology class is a
disconnected multi curve on the pair of pants whilst it is a single connected
simple curve on the torus.

######  Theorem 2.1.

Let $X$ be a possibly incomplete hyperbolic structure on the three holed
sphere. Then the hyperbolic length function extends to a piecewise concave
function on $\mathbb{R}^{2}$.

This extension vanishes

  1. (1)

at just the origin if $X$ has no conical singularity nor puncture

  2. (2)

on a finite number of lines corresponding to simple loops round punctures if
$X$ has punctures but no conical singularity.

  3. (3)

a single line through the origin if $X$ has a single conical singularity but
no cusps.

  4. (4)

countably many lines through the origin if $X$ has a single conical
singularity and at least one cusp.

Further if $X$ has a cone singularity commensurable with $2\pi$ then the
extension is invariant by a transvection.

The points (1) and (2) follow from lower bounds for the length of a geodesic
which can be deduced from convexity. The points (3) and (4) follows from the
fact that an element of $\rho(\mathbb{Z}*\mathbb{Z})$ representing the
holonomy round the cone point is elliptic. Further if the angle at the cone
singularity is commensurable with $2\pi$ then this element is finite order $p$
say. After choosing a loop round the cone singularity as the first element of
a basis of the homology it is easy to show that the level set will be
invariant under:

$(x,y)\mapsto(x,y+px).$

####  2.0.1. Area and counting

As in [11] and [12] it is possible to associate an area to $\ell^{-1}([0,1])$
in a natural way and we study how the area varies with boundary length. The
two plots illustrate the main feature of this variation:

  * •

on the left we see the variation of the level set corresponding to triples
$(2.3,2.3,t)$ as t varies from 4 to $2^{20}$

  * •

on the right we see a plot for $(t,2.3,10)$ as t varies from 2.3 to 2.00243.
We see the spike growing, to accomodate this growth we show only half the
level set and have changed the aspect ratio of the image.

(a)

(b)

Figure 6. Variation of the level sets as one of the boundary length increases
(on the left) or decreases (on the right).

The fact that $\ell^{-1}([0,1])$ may not be compact or even of finite area
means that counting integer points in the associated Markoff morphism orbit is
very different from the punctured torus [11]. Indeed whenever there is a
parabolic element the growth of orbit will be essentially linear. This is easy
to see for the Clebsch cubic from equation (2).

####  2.0.2. Non orientable surfaces

A similar analysis can be carried out for the pair of non orientable surfaces
with fundamental group isomorphic to $\mathbb{Z}*\mathbb{Z}$. The level set of
the length function no longer bounds a convex region and appears to have a
fractal nature.

Figure 7. Part of the level set of the length function of a non orientable
surface.

####  2.0.3. Markoff maps

Following Bowditch, S.P. Tan and his co authors have studied so-called Markoff
maps developping a theory analogous to Conway’s topography for quadratic
forms. Some of our results can be interpreted in terms of this theory. For
example the excursions to infinity of the level sets correspond to the ends of
the Markoff map. We will not however develop the relationship between our
results and Markoff maps here .

##  3\. Automorphisms of $\mathbb{Z}*\mathbb{Z}$

Let $\mathbb{Z}*\mathbb{Z}$ denote the free group on 2 generators which we
will denote $\alpha$ and $\beta$. We think of $\alpha$ and $\beta$ as simple
loops meeting in a single point in a holed torus. An element
$\gamma\in\mathbb{Z}*\mathbb{Z}$ is called primitive, if there exists an
automorphism $\phi$, such that $\gamma=\phi(\alpha)$. If $\phi(\delta)=\beta$,
then $\gamma$ and $\delta$ are called associated primitives.

Following [11] let $\phi:\mathbb{Z}*\mathbb{Z}\rightarrow\mathbb{Z}^{2}$ be
the canonical abelianizing homomorphism. The kernel of $\phi$ is a
characteristic subgroup that is it is
$\text{Aut}(\mathbb{Z}*\mathbb{Z})$-invariant so there is a homorphism from
$\text{Aut}(\mathbb{Z}*\mathbb{Z})$ to the automorphisms of $\mathbb{Z}^{2}$
namely $\mathrm{GL}(2,\mathbb{Z})$. In fact this homomorphism is surjective
and the kernel is exactly the inner automorphisms so that it induces an
isomorphism between the group of outer automorphisms of
$\mathbb{Z}*\mathbb{Z}$ and $\mathrm{GL}(2,\mathbb{Z})$.

  1. (1)

If $\gamma$ is primitive then $\phi(\gamma)$ is a primitive element of
$\mathbb{Z}^{2}$.

  2. (2)

$\text{Aut}(\mathbb{Z}*\mathbb{Z})$ acts transitively on primitive elements of
$\mathbb{Z}*\mathbb{Z}$ and $\mathrm{GL}(2,\mathbb{Z})$ acts transitively on
primitive elements of $\mathbb{Z}^{2}$.

###  3.1. $\mathbb{Z}*\mathbb{Z}$ as a surface group

The group $\mathbb{Z}*\mathbb{Z}$ is isomorphic to the fundamental group of
exactly 4 surfaces, namely

  * •

the three holed sphere

  * •

the one holed mobius band (the two holed projective plane)

  * •

the one holed klein bottle

  * •

the one holed torus

Figure 8. A pair of tripods

It is convenient to think of a topological surface with boundary as a
trivalent ”fat graph”. A fat graph is a finite connected graph equipped with a
cyclic ordering on the half edges incident to each vertex. To fat graph, one
can associate an oriented surface with boundary by replacing edges by thin
oriented rectangles (ribbons) and vertices by disks, and pasting rectangles to
disks according to the chosen cyclic orders at the vertices.

In practice a fat graph is typically trivalent and, if the fundamental group
of the surface is $\mathbb{Z}*\mathbb{Z}$ it will have two vertices and three
edges. If we split each of these edges at its midpoint we have six half edges
each adjacent to exactly one of the vertices. A vertex with its three half
edges is called a tripod. Each of the four surfaces in our list above can be
obtained as ”twisted fat graphs” from a pair of congruent ”fat tripods” as
follows. A fat tripod is a regular hexagon which has flaps attached to three
pairwise non adjacent edges (see Figure ). There are essentially four
different ways to glue, or more correctly sew, the tripods together depending
on whether the flaps are twisted once or not at all.

  * •

if all the flaps are untwisted then one obtains a three holed sphere

  * •

if exactly one flap is twisted then one obtains a one holed mobius band

  * •

if exactly two flaps are twisted then one obtains a one holed klein bottle

  * •

if all the flaps are twisted then the result is a one holed torus.

Following Gilman and Keen [7] we call the the three arcs along which the flaps
are joined seams.

Clearly, if $\Sigma$ is one of our four surfaces then one can obtain a convex
hyperbolic structure on $X$ by replacing the tripods with a pair of congruent
right angled hyperbolic hexagons. When $\Sigma$ is a three holed sphere there
are many good expositions of this procedure, see for example [2], [1]. In the
case of the punctured torus one obtains from seams a triple of geodesic arcs
perpendicular to the boundary. These arcs are so-called ortho geodesics and
this has structure has been quite extensively studied by S.P. Tan and his co
authors [14]. The hyperbolic structures on the one holed Mobius band and the
one holed Klein bottle are less well studied but appear in [13] and [8].

###  3.2. Topological realizations of automorphisms of $\mathbb{Z}*\mathbb{Z}$

It is often useful to identify $\mathbb{Z}*\mathbb{Z}$ with the fundamental
group of the holed torus and think of $\text{Out}(\mathbb{Z}*\mathbb{Z})$, the
outer automorphism group of $\mathbb{Z}*\mathbb{Z}$, as being the mapping
class group of the holed torus. The outer automorphism group is generated by
the so-called Nielsen transformations, which either permute the basis
$\alpha,\beta$, or transform it into $\alpha\beta,\beta$, or
$\alpha\beta^{-1},\beta$. Both these latter transformations can be realized
topologically as Dehn twists on the holed torus.

In what follows we will identify $\mathbb{Z}^{2}$ with the homology of the
surface $\Sigma$ and will call the cover $\tilde{\Sigma}\rightarrow\Sigma$
corresponding to $\ker\phi$ the homology cover or maximal abelian of $\Sigma$.

Figure 9. Maximal abelian cover for the three holed sphere. The boundary loops
lift to the grey curves, the patches appear triangular but in fact are
hexagons joined along sides which are perpendicular to the plane.

There are many fine expositions of the action of
$\text{Out}(\mathbb{Z}*\mathbb{Z})$ on the maximal abelian cover of the one
holed torus. Unfortunately the action does not lift well to the maximal
abelian cover for the three holed sphere. However, we will use a “common
spine” to transfer information.

###  3.3. Spines and retractions

Recall that a loop $\gamma$ is primitive if there is an automorphism $\phi$ of
the free group such that $\phi^{-1}(\gamma)$ is a generator (boundary loop).
It is well known that if $\alpha$ is a primitive loop on a punctured torus
then the unique geodesic freely homotopic to $\alpha$ passes through exactly
two of the three Weierstrass points. Recall that a Weierstrass point is a
fixed point of the elliptic involution. Our reasoning depends on transferring
other well-known results for primitive geodesics on the punctured torus to
primitive geodesics a three holed sphere (pants). This is possible because the
punctured torus and the three holed sphere have spines which are isomorphic -
we think of them as having a common spine.

A spine for a surface is a 1-complex which is a strong deformation retract,
the dotted lines in Figure (8) will glue together to make a spine for the
surface. Epstein and Bowditch used hyperbolic geometry to construct spines of
surfaces with boundary or cusps. The spine for a pair of pants consists of all
the points $x$ in the complement such that the distance to the boundary is
realised by at least two geodesic arcs. To construct a spine for the punctured
torus it is a little more complicated: one first chooses a cusp region $H$
then the spine consists of all the points $x$ in the complement of $H$ such
that the distance $d(x,H)$ is realised by at least two geodesic arcs. It is
easy to see that this does not depend on the choice of $H$. It is easy to see
that the spine for a pair of pants is always a trivalent graph with exactly
two vertices. For a generic punctured torus this also the case.

We will use a spine for a punctured torus which is the union of two simple
loops which generate the fundamental group. For the pair of pants our spine
will be a figure eight curve.

Figure 10. Spines for surfaces. A pair of generating loops in a one holed
torus and a figure eight curve in a pair of pants.

##  4\. Non vanishing and lower bounds for length

We begin by studying the case when there are no punctures. The pants has three
boundary components of lengths
$\ell_{\alpha},\ell_{\beta},\ell_{\alpha\beta^{-1}}$. Given a curve $\gamma$
such that

$\gamma=m\alpha+n\beta,$

with $m,n$ non negative integers then it is possible to do surgery at self
intersection points of $\gamma$ to obtain a simple multi curve each of whose
components is homotopic to either $\alpha$ or $\beta$. This procedure reduces
length and so

$\ell_{\gamma}\geq m\ell_{\alpha}+n\ell_{\beta},$

Suppose that the surface has a single puncture so that, with the notation
above, $\ell_{\alpha}=0$. We think of the surface as being obtained from an
essential sub surface of the thrice punctured sphere. That is $X\subset
Y:=\mathbb{H}/\Gamma(2)$ (where as usual $\Gamma(2)$ is the principal
congruence subgroup) so that $X$ inherits its complex structure from the
inclusion and a hyperbolic structure via the Poincaré metric. By the Ahlfors-
Pick-Schwarz Lemma the length of $\gamma\subset X$ is bounded below by the
length of $\gamma\subset Y$ so it suffices to bound the length on the thrice
punctured sphere.

The length of $\gamma$ is obtained from the trace of a primitive element of
the fundamental group which we denote $C$. We choose generators for the
fundamental group which are a pair of loops round $\alpha$ and $\beta$ denoted
$A$ and $B$ respectively. Gilman [6] has made a study of the word in $A,B$
that represents $C$. Relabelling boundaries if necessary we may assume $m>n$
and then this word takes the form

$C=A^{m_{1}}B\ldots A^{m_{r}}B$

where the $m_{i}\in\\{[m/n],[m/n]+1\\}$. One can then argue that the length of
$\gamma$ is approximately

$n\times\ell_{\gamma_{1}}$

where $\gamma_{1}$ is the loop representing the homotopy class of
$A^{m_{1}}B$. The length of the $\gamma_{1}$ can be calulated from the trace
of the matrix

$\begin{pmatrix}1&2\\\ 0&1\end{pmatrix}^{m_{1}}\begin{pmatrix}1&0\\\
2&1\end{pmatrix}=\begin{pmatrix}1+4m_{1}&2m_{1}\\\
2&1\end{pmatrix}\in\Gamma(2).$

so that $\cosh\ell_{X}(\gamma_{1})/2=2m_{1}+1,$ and

$\ell_{X}(\gamma_{1})\geq 2n\log(2m_{1}).$

It is not difficult to see that if $m>n$ and $m/n\leq K$, so that $n\geq m/K$,
then

$2n\log\left(2\left[\frac{m}{n}\right]\right)\geq\frac{\log(2)}{K}m.$

We have plotted the graph of this estimate and the graph of the catual length
in Figure 11. This is enough to show the non vanishing of the extension as in
(2) of Theorem 2.1.

Figure 11. Hyperbolic length versus our estimate.

##  5\. Geodesic length and convexity for the holed torus

###  5.1. Convexity for the holed torus

Let $\gamma\in\mathbb{Z}*\mathbb{Z}$ be some closed curve and
$\ell_{X}(\gamma)$ the minimum over the lengths wrt the hyperbolic structure
$X$ of all closed curves in the homotopy class of $\gamma$. In fact the
minimum is attained and is lthe length of the unique (oriented) closed
geodesic in the homotopy class.

######  Theorem 5.1.

The length function is convex on $H_{1}(X,\mathbb{Q})$ and so extends
continuously (as a convex function) to $H_{1}(X,\mathbb{R})$.

Since the proof is short and instructive we shall reproduce it now.

Let $X$ be a holed torus equipped with a hyperbolic structure. Then, the
shortest multicurve representing a non-trivial homology class $h$ is a simple
closed geodesic if $h$ is a primitive homology class (that is, not a multiple
of another class), and a multiply covered geodesic otherwise. Further, the
shortest multicurve representing h is unique.

Begin by defining a valuation $\ell$ on the first homology with integral
coefficients, where $\ell(h)$ is just the length of the shortest multicurve
representing h. The valuation of the trivial homology class is defined to be
0. By the previous paragraph $\ell$ statisfies

$\ell(nh)=n\ell(h),\forall n\in\mathbb{N}.$

Moreover if $h,g$ are not commensurable then $\ell$ satisfies the strict
triangle inequality

(3) $\ell(h+g)<\ell(h)+\ell(g).$

This is because the union of the shortest multicurves representing h and g is
not embedded, that is there is a transverse intersection somewhere. Hence, the
shortest multicurve representing h + g is strictly shorter than the union.

##  6\. Concavity for the three holed sphere

To prove concavity we have to prove that, in a certain sense, the opposite of
the triangle holds that is:

(4) $\ell(h+g)>\ell(h)+\ell(g).$

Whilst the triangle inequality followed as a direct consequence of the fact
that a multiple of a simple geodesic in the class $h+g$ was the unique
minimiser, the inequality (4) is be more difficult to prove. Firstly, it is
not true for any choice of $h$ and $g$ and this is why we can only prove
concavity piecewise. Secondly, although our proof is also based on eliminating
transverse intersections to reduce length the argument is more delicate.

So we begin by showing that (4) is true for a few very simple configurations.
We will then give an induction argument to show that it is true in general.

###  6.1. Eliminating intersections

The key point in proving (3) is the fact that the multi geodesics representing
$h$ and $g$ had a transverse intersection somewhere that could be eliminated,
or surgered out, to find a shorter representative for $h+g$. This is iterated
and must terminate when there are no longer any transverse intersections.

Given $h$ and $g$ there are multi geodesic representatives on the three holed
sphere but, unless they are boundary loops, they are no longer simple.
Typically they have self intersections on the seams of the three holed sphere
that Gilman and Keen [5][7] call essential intersections. In a certain sense
the opposite of what happened in the holed torus occurs. Beginning with
$k\gamma$ the multi geodesic representing $h+g$ and we eliminate an essential
intersection to get a pair of multi geodesics which have (fewer transverse
intersections) smaller total length than $\ell(k\gamma)=k\ell(\gamma)$.
Iterating this process we should be able to show the required inequality. For
example the figure eight curve has a single self intersection and after
surgery it gives rise to a pair of curves homotopic to distinct boundary
curves (which are disjoint). In the next section we treat this example more
formally.

###  6.2. The figure eight curve.

Let us begin by considering the triple $\alpha,\beta$ and $\alpha\beta$ where,
as usual $\alpha,\beta,\alpha\beta^{-1}$ are simple loops round the boundary
components and $\alpha\beta$ is a figure of eight curve. The figure of eight
curve has a single essential intersection point and there are two loops
meeting at this point one homotopic to $\alpha$ and the other $\beta$ ( in
fact the surface retracts onto this figure eight curve and we will need this
later.) Clearly since neither of these loops is smooth they are longer than
the closed geodesic in the same homotopy class and since length of
$\alpha\beta^{-1}$ is the sum of these loops we have:

(5) $\ell(\alpha\beta)>\ell(\alpha)+\ell(\beta).$

Let’s take this further consider the pair of loops

$(\alpha\beta)^{n}\beta,(\alpha\beta)^{n}\alpha$

so that in the abelianisation one has

$(n+1)\alpha\beta=(\alpha\beta)^{n}\beta+(\alpha\beta)^{n}\alpha.$

The loops $(\alpha\beta)^{n}\beta$ and $(\alpha\beta)^{n}$ retract onto the
figure eight curve $\gamma$ and by choosing our retraction carefully we may
suppose that this map is of degree $n+1$. Now the loops on the $\gamma$ are
not smooth so they are stricly longer than the geodesics in the the same
homotopy class and so

(6)
$(n+1)\ell(\alpha\beta)>\ell((\alpha\beta)^{n}\beta)+\ell((\alpha\beta)^{n}\alpha).$

We define the positive cone determined by a pair of oriented loops to be the
subset of the homology consisting of linear combinations with positive
coefficients. More generally if we have

$k\alpha\beta=m\gamma_{1}+n\gamma_{2}$

with $\gamma_{1},\gamma_{2}$ primitive loops in the positive cone generated by
the oriented loops $\alpha$ and $\beta$ in the homology then, since the
surface retracts onto the figure eight graph determined by the $\alpha\beta$
each of the loops $\gamma_{i}$ is homotopic to a piecewise geodesic curve on
$\gamma$. So, essentially by the same argument, the reverse triangle
inequality is still true, that is

(7) $k\ell(\alpha\beta)>m\ell(\gamma_{1})+n\ell(\gamma_{2}),$

where $k$ is the degree of the retraction from the multi curve
$m\gamma_{1}\cup n\gamma_{2}$.

###  6.3. Convexity in general

To prove convexity we decompose our loop $\gamma$ into a pair of generator
loops $\alpha^{\prime},\beta^{\prime}$, which are not necessarily boundary
loops, but so that $\gamma$ is still a product

$\displaystyle\gamma=\alpha^{\prime}\beta^{\prime}.$

We do this by splitting at an essential self intersection then writing
$\gamma_{i}$ as words in these. With such a decomposition the reverse triangle
inequality follows since $\alpha$ and $\beta$ minimise length in their
respective homotopy classes by applying the argument of the previous
paragraph.

More formally :

######  Theorem 6.1.

Let $\gamma_{1},\gamma_{2}$ be a pair of primitive geodesic loops such that
there are positive integers $m,n\geq 0$ with

$k\gamma=m\gamma_{1}+n\gamma_{2}.$

in the abelianisation. Then there is a positive integer $r$ such that
$rk\gamma$ can be decomposed into $rm$ loops homotopic to $\gamma_{1}$ and
$rn$ loops homotopic to $\gamma_{2}$

####  6.3.1. Very local concavity

We can prove a version of Theorem 6.1: decomposition for pairs
$\gamma_{1},\gamma_{2}$ in a (non uniformly) small neighborhood of the loop
$\gamma$ and with $c=1$. From this we can deduce concavity on the neighborhood
and by a local to global argument on the whole positive quadrant. By a small
neighborhood of $\gamma$ we mean we mean those primitive $\gamma^{\prime}$
such that $\gamma/\|\gamma\|_{2}$ and
$\gamma^{\prime}/\|\gamma^{\prime}\|_{2}$ are close (where, as usual
$\|.\|_{2}$ denotes the Euclidean norm). In fact, if $\gamma=p\alpha+q\beta$
with $p,q\geq 0$ coprime integers then this neighborhood will be the positive
cone determined by the loops

$\displaystyle a\alpha+c\beta$ $\displaystyle b\alpha+d\beta$

where $a/c$ and $b/d$ are Farey neighbors of each other and $p/q$.

Our argument uses the notion of a staircase curve, that is a curve which is
contained in the grid
$(\mathbb{R}\times(\mathbb{Z}+\frac{1}{2}))\cup((\mathbb{Z}+\frac{1}{2})\times\mathbb{R})\subset\mathbb{R}^{2}$
note that this grid is wholly contained in the punctured plane
$\mathbb{R}^{2}\setminus\mathbb{Z}^{2}$.

Figure 12. Transfering concavity

The punctured plane $\mathbb{R}^{2}\setminus\mathbb{Z}^{2}$ can be identified
with the maximal abelian cover of the punctured torus: if
$\Gamma\simeq\mathbb{Z}^{2}$ denotes the group of deck transformations then
the punctured torus is obtained as a quotient:

$\left(\mathbb{R}^{2}\setminus\mathbb{Z}^{2}\right)/\Gamma.$

There is a $\Gamma$-equivariant retraction $H_{t}$ from the punctured plane to
a grid of horizontal and vertical lines (see Figure (12)). So this grid is in
fact the set of lifts of a spine for the once punctured torus.

Recall that a Weierstrass point on the punctured torus is a fixed point of the
elliptic involution that is the involution of
$\left(\mathbb{R}^{2}\setminus\mathbb{Z}^{2}\right)/\Gamma$ induced by
$\mathbb{R}^{2}\rightarrow\mathbb{R}^{2},\,v\mapsto-v$. It is easy to see that
the set of lifts of these points are respectively
$\mathbb{Z}^{2}+(\frac{1}{2},0)$ , $\mathbb{Z}^{2}+(0,\frac{1}{2})$ and
$\mathbb{Z}^{2}+(\frac{1}{2},\frac{1}{2})$. So the grid contains lifts of the
three Weierstrass points - these occur at the vertices of the grid and the
midpoints of the horizontal and vertical edges. We may also identify the grid
with the lift of a spine for the pants. We quite simply choose a figure eight
curve and a retraction onto it such that each of the three essential
intersections is fixed. Thus we think of the grid as being a “common spine”
for the maximal abelian cover of both the punctured torus and the pair of
pants. This allows us to visualise essential intersections of geodesics as
lifts of Weierstrass points on the grid.

Consider the line

$\hat{\gamma}=\left\\{\begin{pmatrix}t\\\
t+\frac{1}{2}\end{pmatrix},t\in\mathbb{R}\right\\}$

this curve is important as its retraction to the spine of
$H_{1}(\hat{\gamma})$ is a staircase curve which coincides with a lift of the
figure eight curve. The figure eight curve crosses the three seams of the
pants but only one gives rise to an essential self intersection. The
corresponding Weierstrass point is the one at the vertex of the grid. Doing
surgery at each vertex yields the horizontal and vertical lines which are
lifts of curves homotopic to boundary components of the pants. Any other
straight line $\hat{\zeta}_{1}$ avoiding the integer lattice and with positive
gradient retracts onto a “staircase curve” $H_{1}(\hat{\zeta}_{1})$ on the
grid and so decomposes as a sequence of horizontal and vertical edges from
which we obtain the partition as in the previous subsection.

Now consider $p,q>0$ coprime integers. It is well known that $p/q$ is the
descendant of two Farey neighbors that is a pair $a/c,b/d\in\mathbb{Q}_{+}$
with $ad-bc=1$, such that

$\frac{p}{q}=\frac{a+b}{c+d}.$

This means that we can think of $\begin{pmatrix}p\\\ q\end{pmatrix}$ as being
the image of $\begin{pmatrix}1\\\ 1\end{pmatrix}$ under the matrix

$M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{Z}).$

So under the linear transformation $f_{M}:v\mapsto Mv$ our line $\hat{\gamma}$
is mapped to $f_{M}(\hat{\gamma})$ which is a line of gradient $p/q$ avoiding
the half integer lattice. Note that the image of the staircase curve
$H_{1}(\hat{\gamma})$ is not a staircase curve but it is a union of line
segments and homotopic to $f_{M}(\hat{\gamma})$. Now for any
$p^{\prime}/q^{\prime}$ such that

$\frac{a}{c}\leq\frac{p^{\prime}}{q^{\prime}}\leq\frac{b}{d}$

the image under $f_{M}^{-1}$ of the ray through $\begin{pmatrix}p^{\prime}\\\
q^{\prime}\end{pmatrix}$ is a line $\hat{\zeta}$ with positive gradient which
retracts onto the staircase curve $H_{1}(\hat{\zeta})$ as in the previous
paragraph. Thus $\zeta$ decomposes as a sequence lifts of sub loops of the
figure eight curve. Finally $f_{M}$ takes $\hat{\zeta}$ to a line of gradient
$p^{\prime}/q^{\prime}$ and $H_{1}(\hat{\zeta})$ to a curve which is a union
of line segments between lifts of Weierstrass points/essential intersections
and so they yield the decomposition of the loop $f_{M}(\zeta)$ as subloops of
the loop determined by $f_{M}(\gamma)$ on the pair of pants.

This procedure is illustrated for $p=3,q=2$ in Figure (12).

Finally there is some accounting to be done if $p^{\prime}/q^{\prime}$ and
$p^{\prime\prime}/q^{\prime\prime}$ are in the small neighborhood bounded by
the Farey ancestors of $p/q$ then there exists $m,n$ such that

$k\begin{pmatrix}p\\\ q\end{pmatrix}=m\begin{pmatrix}p^{\prime}\\\
q^{\prime}\end{pmatrix}+n\begin{pmatrix}p^{\prime\prime}\\\
q^{\prime\prime}\end{pmatrix}$

where $k$ is the determinant of

$\begin{pmatrix}p^{\prime}&p^{\prime\prime}\\\
q^{\prime}&q^{\prime\prime}\end{pmatrix},$

and this proves the theorem at least locally.

####  6.3.2. Bigger neighborhoods

Now we can expand the convexity to a bigger neighborhood by inheriting the
neighborhoods of the Farey ancestors $a/c$ and $b/d$ of $p/q$. By the
preceding paragraph there is a neighborhood of $a/c$ for which each curve
retracts onto a sequence of sub loops of the corresponding closed geodesic.
Again by the preceding paragraph each of these sub loops contracts onto a sub
loop of the closed geodesic associated to $p/q$.

Iterating this process, and keeping track of the “accounting” one can prove
the theorem on the whole of the positive quadrant.

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