# Fourier decay, Green’s functions and Schottky groups

Ilyas Bayramov

(August 2023)

###### Abstract

In this note, I would like to discuss an approach to the construction of
Green’s function of laplacians on algebraic surfaces, indicated by Manin in
[9], towards the computation of the Green’s function on surfaces using
Schottky uniformization. We shall see that the exact geometric interpretation
of the formula mentioned there is obscure, and try to remedy the situation by
investigating convergence of deformations of that formula.

##  1 Acknowledgments

I thank Peter Kosenko and Dennis Sullivan for many useful suggestions.

##  2 Motivation for Manin’s formula

There is a certain generalization due to Arakelov of the usual intersection
pairing between the divisors that is useful, for example, in the proof of
Mordell’s theorem. Here by divisor we mean a Weil divisor, i.e., a complex
linear combination of points on the Riemann surface $S$.

To each divisor $a$ of degree $0$ on a (smooth) algebraic curve $S$, by
Riemann-Roch [5], corresponds a meromorphic differential form $\omega_{a}$
such that

$a=\operatornamewithlimits{\sum}\limits_{x\in S}res_{x}(\omega_{a})\cdot x.$

This differential form is unique up to an addition of a holomorphic
differential form to it. Since the real parts of periods of a holomorphic
differential determine it uniquely (by, say, Riemann period relations) and we
have the freedom to choose any holomorphic differential, we can insist that
$\omega_{a}$ has all of its periods purely imaginary.

Then $\text{d}g_{a}:=\omega_{a}-\bar{\omega}_{a}$ is exact since all of its
periods are $0.$ $g_{a}$ is then called the Green-Arakelov function.

Definition[5]: Let $b=\operatornamewithlimits{\sum}\limits_{y}m_{y}(y),$ a
divisor of degree $0$; let $a$ be another such divisor,
$|a|\cap|b|=\emptyset$, $(y)\in S,$ $S$ an algebraic curve. Then the height
pairing between $a$ and $b$ is
$g(a,b)=\operatornamewithlimits{\sum}\limits_{y}m_{y}g_{a}(y).$

The goal is to compute $g_{a}.$ To do that, we need to take a detour into low-
dimensional topology.

##  3 Schottky uniformization: existence

Schottky uniformization of a Riemann surface $S$ is a representation
$\rho:\pi_{1}(S)\to PSL(2,\mathbb{C}),$ satiasfying the following conditions.
Its image, called a Schottky group, is a free group $\Gamma$ that is purely
hyperbolic, i.e., the absolute values of traces of its elements are strictly
bounded by $2$ from below. $PSL(2,\mathbb{C})$ acts on $\mathbb{H}^{3}$ by
isometries, and on $S^{2}$ by linear fractional transformations. $S^{2}$ can
be identified with the boundary of $\mathbb{H}^{3}$ at infinity (aka ”the
conformal infinity”).

Now, any infinite discrete group acting on a compact space, in this case
$\mathbb{H}^{3}\cup S^{2},$ must have accumulation points. The set of such
accumulation points must then be contained in $S^{2}$ (because
$PSL(2,\mathbb{C})$ acts on $\mathbb{H}^{3}$ by isometries). For a given
$\Gamma$ we denote it by $\Lambda(\Gamma)$ and call it the limit set. Its
complement in $\S^{2}$ is denoted by $\Omega(\Gamma)$ and is called the domain
of discontinuity. By Sullivan’s dictionary, these correspond to Julia and
Fatou sets, respectively.

”Uniformization” then comes from the application of Ahlfors’ finiteness
theorem.

It states that for any finitely generated discrete subgroup of
$PSL(2,\mathbb{C})$,

$\Omega(\Gamma)/\Gamma$

is a Riemann surface of finite area. Hence, in this particular case it is, as
well, and to each Riemann surface we can associate a corresponding Schottky
uniformization by Koebe’s retrosection theorem.

##  4 Schottky uniformization: uniqueness

This uniformization is not unique; to see where exactly the non-uniqueness
arises from, we must consider the action of a Schottky group $\Gamma$ on all
of $\mathbb{H}^{3}\cup\Omega(\Gamma).$ Then the quotient $K(\Gamma)$ of such
an action admits a hyperbolic metric in its interior, and it is actually
evident that $K(\Gamma)$ is a handlebody. Then $S:=\Omega(\Gamma)/\Gamma$
determines a point in the Teichmüller space up to the action of the elements
of the mapping class group that have representatives extending to
homeomorphisms of $K(\Gamma)$ that are isotopic to the identity. This subgroup
of $\text{Mod}(S)$ is denoted by $\text{Mod}_{0}(S)$ and is known as the
handlebody group of a fixed genus $g.$

##  5 Compactification $\bar{\mathfrak{T}}$ of the Teichmuller space

Thurston defined a compactification of the Teichmuller space as follows:
consider the set $\mathcal{S}$ of isotopy classes of simple closed curves on a
surface. Then for two such classes $a,b\in\mathcal{S}$ the geometric
intersection number $i(a,b)$ is given by the smallest number of intersections
between the representatives of $a$ and $b$, respectively. Denote by
$\mathbb{R}_{+}$ the set of nonegative reals with the usual induced topology.
Then there exists a map

$i_{*}:\mathcal{S}\to\mathbb{R}_{+}^{\mathcal{S}}\backslash 0,$
$i_{*}(a)(b)=i(a,b).$

Now, there also exists a map $\pi:\mathbb{R}_{+}^{\mathcal{S}}\backslash 0\to
P(\mathbb{R}_{+}{\mathcal{S}}).$ Here arises the question of topology on these
spaces: we take $\mathbb{R}_{+}^{\mathcal{S}}$ with the weak (product)
topology, and $P(\mathbb{R}_{+}^{\mathcal{S}})$ with the quotient topology.
Then the closure $\bar{\mathcal{S}}$ of $\pi\circ i_{*}(\mathcal{S})$ in
quotient topology is homeomorphic to a sphere $S^{6g-6-2b},$ where $b$ is the
number of punctures.

On the other hand, there also exists an embedding of Teichmuller space into
$P(\mathbb{R}^{\mathcal{S}}).$ It is given by the length function

$l:\mathfrak{T}\times\mathcal{S}\to\mathbb{R}_{+}^{\mathcal{S}},$
$l(\theta,\alpha)=\inf\limits_{\gamma\in\alpha}\theta(\gamma),$

where $\gamma$ is a simple closed curve in the isotopy class of $\alpha.$
Since this is well-defined, because both $\theta$ and $\alpha$ are defined up
to isotopy, this gives an embedding
$l_{*}:\mathfrak{T}\to\mathbb{R}_{+}^{\mathcal{S}}\backslash 0$ which is a
homeomorphism onto the image.

Now it is obvious that the geometric intersection number of $a\in\mathcal{S}$
with itself is $0,$ and by [2], the geometric intersection number can be
extended to $\mathfrak{T}$ and $\bar{\mathcal{S}}$ in such a way that the
self-intersection number is respectively a non-zero constant on all of
$l_{*}(\mathfrak{T})$ and $0$ on $\bar{\mathcal{S}}.$ Since we consider the
weak topology on $\mathbb{R}_{+}^{\mathcal{S}}\backslash 0,$ we see that if a
sequence $\theta_{n}$ of elements of $l_{*}(\mathfrak{T})$ leaves any compact
set of $\mathbb{R}_{+}^{\mathcal{S}}\backslash 0$ in such topology, then we
must have that for at least one of the elements $\alpha$ of $\mathcal{S}$
$l(\theta_{n},\alpha)\to\infty.$ Then if a sequence $\pi(\theta_{n})$
converges to some point $b$ in $P(\mathbb{R}^{\mathcal{S}}),$ a sequence of
their preimages in $\mathbb{R}_{+}^{\mathcal{S}}$ must converges to a function
$a\in\pi^{-1}(b);$ that means that there exists numbers $\lambda_{n}>0$ such
that $\lambda_{n}\theta_{n}$ converges to $a;$ as per the above,
$\lambda_{n}\to 0,$ as at least one of the coordinates
$(\theta_{n}(s))_{s\in\mathcal{S}}$ goes to $\infty.$ Then by the bilinearity
(!) of the intersection number
$i(\lambda_{n}\theta_{n},\lambda_{n}\theta_{n})=|\lambda_{n}|^{2}i(\theta_{n},\theta_{n})=C|\lambda_{n}|^{2}\to
0=i(a,a).$ This implies that $a\in\bar{\mathcal{S}},$ by the converse of the
above statement about $\bar{\mathcal{S}},$ which is less obvious. Notice that
this argument is very similar to a measure concentration phenomenon. Thus,
$\bar{\mathfrak{T}}=\pi\circ l_{*}(\mathfrak{T})\cup{\mathcal{S}}$ is a
compactification of $\mathfrak{T}.$

##  6 The action of $\text{Mod}_{0}(S)$ on $\bar{\mathfrak{T}}$

Profoundly, the situation in this case is reminiscent of the Kleinian group
action on $\mathbb{H}^{3}:$ for the subgroup $\text{Mod}(H)$ of
$\text{Mod}(S)$ that consists of elements that have representatives extending
to diffeomorphisms of the handlebody $H,\partial H=S$, there is a domain of
discontinuity on $\bar{\mathfrak{T}},$ and a limit set that is contained in
$\bar{\mathcal{S}}.$ Now, following Masur [10], lemma 1.1, we have that if a
curve on $S$ bounds a disk in the corresponding handlebody, and
$\alpha_{1},\dots,\alpha_{g}$ are simple closed curves that bound disks in $H$
and cut $S$ into a $2g$ holed sphere, then $\beta\in\mathcal{S}$ bounds a disk
iff $i(\beta,\alpha_{i})=0$ or we have the following situation. Suppose
$i(\beta,\alpha_{i})\neq 0$ for some $i$. Choose
$(\beta_{l})_{l\in\\{1,\dots,g\\}}\in\mathcal{S}$ with
$i(\beta_{l},\alpha_{j})=\delta_{lj}$ and $\beta_{l}-$free generators of
$\pi_{1}(H)=\Gamma.$ As a word in $\beta_{l}$ and $\beta_{l}^{-1},\beta$ is
trivial. Up to a cyclic permutation, there must be either
$\beta_{j}\beta_{j}^{-1}$ or $\beta_{j}^{-1}\beta_{j}$ somewhere in the word.
But then $\beta$ crosses $\alpha_{j}$ and then crosses it in the opposite
direction.

Now, consider $\mathcal{S}_{c}\subset\mathcal{S},$ the set of simple closed
curves bounding disks in the handlebody. Then by Theorem 2.2 in Masur
$\bar{\mathcal{S}_{c}}$ is the limit of the actions of both $\text{Mod}(H)$
and $\text{Mod}_{0}(S).$

Finally, there is the canonical measure $m$ on $\bar{\mathcal{S}},$ see
Masur’s theorem $B,$ such that
$m(N:=\\{a\in\bar{\mathcal{S}}-\bar{\mathcal{S}_{c}}|i(a,b)=0\text{ for some
}b\in\bar{\mathcal{S}_{c}}\\})=0.$ Taking
$\mathcal{O}:=(\bar{\mathcal{S}}-\bar{\mathcal{S}_{c}})-N,$ we have that
$\mathcal{O}$ is the domain of discontinuity of the action of both
$\text{Mod}(H)$ and $\text{Mod}_{0}(S)$ on $\mathcal{S}$ by theorem 2.1 in
Masur. Moreover, in genus $2,$ according to Masur, the limit set
$\bar{\mathcal{S}_{c}}$ has measure $0.$

##  7 Deforming the Schottky group

We consider here $AH(H)$ to be the set of all discrete faithful
representations of $\Gamma$ into $PSL(2,\mathbb{C})$ up to conjugation. Now,
according to Li-Ohshika-Lecuire [12], we have that if $\rho_{n}\in AH(H)$
corresponds to a sequence of elements of $\mathfrak{T}$ that converges to an
element of Masur domain, then $\rho_{n}$ converges to an element of $AH(H).$
Finally, we have, according to Anderson-Canary-Culler-Shalen [1], that the set
of Kleinian groups with the limit set the whole sphere is dense in the
boundary of the domain of convex-cocompact representations, i.e., in this
case, the quasi-conformal deformations of a Schottky group. Thus, if a
sequence of elements $a_{n}$ of $\mathfrak{T}$ leaves any compact subset of
$\mathfrak{T}$ in the quotient of weak topology described above, then its
limit $a\in\mathfrak{T}$ almost always corresponds to a group $\Gamma_{a}$
with limit set $\Lambda(\Gamma_{a})=S^{2}.$ This is why we can almost
quasiconformally deform a Schottky group to another one that has $S^{2}$ as
its limit set.

##  8 Manin’s formula

In what follows, $\langle a,b,c,d\rangle$ denotes the cross-ratio of
$a,b,c,d.$

Using Schottky uniformization, Manin [9] defined $\omega_{a}$ for any divisor
of degree $0$, as in section $1$; for $a=(x)-(y)$ it is

$\nu_{(x)-(y)}-\operatornamewithlimits{\sum}\limits_{l}X(x,y)\omega_{l},$

where

$\nu_{(x)-(y)}=\text{dlog}W_{(x)-(y),z_{0}}(z)$

,

$X_{l}(a,b)=-\sqrt{-1}\frac{\operatornamewithlimits{\sum}\limits_{h\in
S(g_{k})}\text{log}|\langle
a,b,z^{+}(h),z^{-}(h)\rangle|}{\operatornamewithlimits{\sum}\limits_{h\in
C(g_{k}|g_{l})}\text{log}|\langle
z^{+}(g_{k})z^{-}(g_{k}),hz^{+}(g_{l}),hz^{-}(g_{l})\rangle|},$
$\omega_{l}=-\sqrt{-1}\frac{1}{2\pi}\text{dlog}W_{(g_{k}z_{1})-(z_{1}),z_{0}}(z).$

Here,

$W_{d,q}(r)=\operatornamewithlimits{\prod}\limits_{h\in\Gamma}\frac{w_{d}(hr)}{w_{d}(hq)},$

for $d$ a divisor in $\mathbb{C}P^{1},$ $w_{d}$ is the meromorphic function
with this divisor. Also, $S(g_{k})$ is the conjugacy class of $g,$ and
$C(g_{k}|g_{l})-$double coset representatives.

##  9 Simplification of Manin’s formula

It is known as a consequence of Fay’s trisecant theorem [14] that

$\text{exp}(\operatornamewithlimits{\int}\limits_{a_{1}}^{a_{2}}\omega_{(x)-(y)})=\frac{\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{1}}^{x}\overrightarrow{\omega})\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{2}}^{y}\overrightarrow{\omega})}{\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{2}}^{x}\overrightarrow{\omega})\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{1}}^{y}\overrightarrow{\omega})},$
$Re\operatornamewithlimits{\int}\limits_{a_{1}}^{a_{2}}\omega_{(x)-(y)}=\log(\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{1}}^{x}\overrightarrow{\omega}))+\log(\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{2}}^{y}\overrightarrow{\omega}))$
$-\log(\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{2}}^{x}\overrightarrow{\omega}))-\log(\theta(\alpha+\operatornamewithlimits{\int}\limits_{a_{1}}^{y}\overrightarrow{\omega}));$

then by Roelcke and Fay [7] we have

$\log(\theta(\alpha+\operatornamewithlimits{\int}\limits_{x}^{y}\overrightarrow{\omega}))=\int{dS(p)}\int{dS(q)}Re\operatornamewithlimits{\int}\limits_{x}^{p}\omega_{(y)-(q)},$

where

$Re\operatornamewithlimits{\int}\limits_{x}^{p}\omega_{(y)-(q)}=\operatornamewithlimits{\sum}\limits_{h\in\Gamma}\log|\langle
y,q,hx,hp\rangle|;$

so we have as the Green’s function

$\operatornamewithlimits{\int}\limits_{\Omega(\Gamma)}\operatornamewithlimits{\int}\limits_{\Omega(\Gamma)}Re\operatornamewithlimits{\int}\limits_{x}^{p}\omega_{(y)-(q)}dS(p)dS(q)$

where $dS$ is the $\Gamma-$invariant measure on $\Omega(\Gamma).$

##  10 Fixed-point theorems

We would like to rephrase the existence of the solution to Laplace’s equation
on $\Lambda(\Gamma)$ with potential (which is equivalent to being able to find
Green’s function) through a fixed point problem. We follow Evans [4] in this.
Consider the following setup: assume that the smooth functions
$\\{w_{k}\\}_{k\in\mathbb{N}}$ form an orthonormal basis of $W^{1,2}_{0}$,
where $W^{1,2}_{0}$ is the closure of the space of compactly supported smooth
functions in $W^{1,2},$ the Sobolev space of square-integrable functions with
square-integrable weak derivative. For example, we can take $w_{k}$ to be
harmonics so that the corresponding coefficients of $w_{k}$ in a series
converging to the function $f\in W^{1,2}_{0}$ are Foruier coefficients.
Consider the system of equations

$\operatornamewithlimits{\int}\limits_{U}\mathbf{a}(Du_{m})\cdot
Dw_{k}dx=\operatornamewithlimits{\int}\limits_{U}fw_{k}dx,$

where $u_{m}=\operatornamewithlimits{\sum}\limits_{k=1}^{m}d^{k}_{m}w_{k},$
but for each $u_{m}$ we only consider those equations up to $k=m.$ Here
$\mathbf{a}:\mathbb{R}^{n}\to\mathbb{R}^{n},\mathbf{a}=\\{a^{1},\dots,a^{n}\\}$
is a vector field. Also, we must impose the following on $\mathbf{a}:$

$(\mathbf{a}(p)-\mathbf{a}(q))(p-q)\geq 0,$ $\mathbf{a}\leq C(1+|p|),$
$\mathbf{a}(p)\cdot p\geq\alpha|p|^{2}-\beta$

for some $C,\alpha>0$ and $\beta\geq 0.$ Then we can use the fixed point
theorem to show that, if we consider

$v^{k}(d):=\operatornamewithlimits{\int}\limits_{U}\mathbf{a}(\operatornamewithlimits{\sum}\limits_{j=1}^{m}d_{j}Dw_{j})\cdot
Dw_{k}-fw_{k}dx(k=1,\dots,m),$

over all $d=(d_{1},\dots,d_{m})\in\mathbb{R}^{m},$ then $v^{k}(d)$ has a zero.

This equation is equivalent to Laplace’s equation $-\Delta u+cu=0,$ where
$c=\frac{2a\Delta a-(\nabla a)^{2}}{4a^{2}}$ (see [6], for example).

We can interpret the space of $d^{\prime}$s in this case as a space of Fourier
coefficients. Naud et al. [8] have established (non-uniform) boundedness of
Fourier coefficients of functions on the limit set of Schottky groups. Thus,
we have that the Green’s function exists on $\Lambda(\Gamma)$ for any $\Gamma$
free convex-cocompact.

Now, by [11], we have that the limit sets of algebraically convergent sequence
of Schottky groups, defined in section 6, converge to the limit set of the of
the group $\Gamma_{a}$ in Gromov-Hausdorff sense. On the other hand, following
Quint [15], we have that the Patterson-Sullivan measure is in the Lebesgue
class of measures on $S^{2},$ hence it is doubling with respect to the Fubini-
Study metric on $S^{2},$ which we identify with $\mathbb{C}P^{1}$. By the
results of [3] adapted to the case of sets, rather than manifolds, we have
that if we have a sequence of metric spaces equipped with measures that are
doubling with respect to those metrics, then the Green’s functions on those
sets converge to a Green’s function on the Gromov-Hausdorff limit of those
sets.

Finally, considering both

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\log|\langle
y,q,hx,hp\rangle|$

and

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a_{n}}}\log|\langle
y,q,hx,hp\rangle|,$

$\Gamma_{a}$ being the algebraic limit of $\Gamma_{a_{n}}$ as above, we have

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\log|\langle
y,q,hx,hp\rangle|=$
$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\chi_{\Omega(\Gamma_{a_{n}})}(\langle
y,q,hx,hp\rangle)\log|\langle y,q,hx,hp\rangle|+$
$+\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\chi_{\Lambda(\Gamma_{a_{n}})}(\langle
y,q,hx,hp\rangle)\log|\langle y,q,hx,hp\rangle|,$

where $\chi^{\prime}\text{s}$ are indicator functions. Since the action of
$\Gamma_{a}$ is topologically dense on $S^{2},$ that is, the closure of any
orbit is dense, we have that, in particular, any orbit of $\Gamma_{a_{n}}$
that lies in $\Omega(\Gamma_{a_{n}})$ can be approximated by the orbit of the
same point under $\Gamma_{a}.$ Since the action of $\Gamma_{a_{n}}$ on
$\Lambda(\Gamma_{a_{n}})$ is also topologically dense, we can estimate

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\chi_{\Lambda(\Gamma_{a_{n}})}(\langle
y,q,hx,hp\rangle)\log|\langle y,q,hx,hp\rangle|$

by

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a_{n}}}\chi_{\Lambda(\Gamma_{a_{n}})}(\langle
y,q,hx,hp\rangle)\log|\langle y,q,hx,hp\rangle|,$

which converges by [8].

Thus, we have the following:

Theorem: For almost any Schottky representation $\Gamma_{0}$ of a fixed free
group, for almost any point at the boundary of the Schottky space, there
exists an unbounded sequence of representations $\Gamma_{n}$ converging to it
and quasiconformal maps on $S^{2}_{\infty},$ conjugating the action of
$\Gamma_{0}$ to $\Gamma_{n},$ such that for any $n$, the expression
$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a_{n}}}\log|\langle
y,q,hx,hp\rangle|$, where $\Gamma$ is the limit of $\Gamma_{n}$ in $AH(S),$
for any quadruple of points $y,q,x,p$, converges.

Proof: If $\langle y,q,x,p\rangle\in\Lambda(\Gamma_{a_{n}}),$ this follows by
[8] and invariance of $\Lambda(\Gamma_{a_{n}})$. If $\rangle
y,q,x,p\langle\in\Omega(\Gamma_{a_{n}}),$ then we can estimate it from above
by

$\operatornamewithlimits{\sum}\limits_{h\in\Gamma_{a}}\chi_{\Omega(\Gamma_{a_{n}})}(\langle
y,q,hx,hp\rangle)\log|\langle y,q,hx,hp\rangle|,$

which converges by the above. This concludes the proof.

Remark: The result above generalizes [13].

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