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% Author Package file for use with AMS-LaTeX 1.2
\controldates{27-MAR-2000,27-MAR-2000,27-MAR-2000,27-MAR-2000}
 
\documentclass{era-l}
\usepackage{graphics}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newcounter{bean1}
\newcounter{bean2}

\issueinfo{6}{03}{}{2000}
\dateposted{March 28, 2000}
\pagespan{21}{30}
\PII{S 1079-6762(00)00076-7}
\def\copyrightyear{2000}
\copyrightinfo{2000}{American Mathematical Society}

\begin{document}
\title[Mating quadratic maps with
Kleinian groups]{Mating quadratic maps with Kleinian groups via
quasiconformal surgery}

\author{S. R. Bullett}
\address{School of Mathematical Sciences, Queen Mary and Westfield
College, University of London, Mile End Road, London E1 4NS,
United Kingdom}
\email{s.r.bullett@qmw.ac.uk}

\author{W. J. Harvey}
\address{Department of Mathematics, King's College, University of London,
Strand, London WC2R 2LS, United Kingdom}
\email{bill.harvey@kcl.ac.uk}

\commby{Svetlana Katok}

\date{December 22, 1999}

\subjclass[2000]{Primary 37F05; Secondary 30D05, 30F40, 37F30}

\keywords{Holomorphic dynamics, quadratic maps, Kleinian groups,
quasiconformal surgery, holomorphic correspondences}

\begin{abstract}
Let $q:\hat{\mathbb C} \to \hat{\mathbb C}$ be any quadratic
polynomial and $r:C_2*C_3 \to PSL(2,{\mathbb C})$ be any faithful
discrete representation of the free product of finite cyclic
groups $C_2$ and $C_3$ (of orders $2$ and $3$) having connected
regular set. We show how the actions of $q$ and $r$ can be
combined, using quasiconformal surgery, to construct a $2:2$
holomorphic correspondence $z \to w$, defined by an algebraic
relation $p(z,w)=0$.
\end{abstract}

\maketitle

\section{Introduction}

Given two abstractly isomorphic Fuchsian groups $G_1\subset
PSL(2,{\mathbb R})$ and $G_2\subset PSL(2,{\mathbb R})$, acting on
the upper and lower halves ${\mathcal U}$ and ${\mathcal L}$ of
the complex plane respectively, each having limit set
$\hat{\mathbb R}={\mathbb R} \cup \infty$, and such that the
action of $G_1$ on $\hat{\mathbb R}$ is {\em topologically}
conjugate to that of $G_2$, it is well known that one can {\em
mate} the actions of $G_1$ and $G_2$ to obtain a Kleinian group $G
\subset PSL(2,{\mathbb C})$, isomorphic as an abstract group to
both $G_1$ and $G_2$, such that the limit set $\Lambda$ of $G$ is
a simple closed (fractal) curve and the actions of $G$ on the two
components of $\Omega={\hat{\mathbb C}}-\Lambda$ are {\em
conformally} conjugate to those of $G_1$ on ${\mathcal U}$ and
$G_2$ on ${\mathcal L}$.

Equally, given two polynomial maps $P$ and $Q$ of the same degree
$n$, having connected filled Julia sets $K(P)$ and $K(Q)$
respectively, it is well known that in certain cases one can {\em
mate} the actions to obtain a {\em rational} map $R$ such that the
complement $\Omega$ of the Julia set $J(R)$ is a disjoint union of
two open sets, on one of which the action of $R$ is conformally
conjugate to that of $P$ on the interior $K(P)^\circ$ of its
filled Julia set, and on the other of which the action of $R$ is
conformally conjugate to that of $Q$ on $K(Q)^\circ$. A necessary
condition for a mating of two quadratic polynomials $P:z \to
z^2+c$ and $Q:z \to z^2+c'$ to exist is that $c$ and $c'$ should
not belong to conjugate limbs of the connectivity locus in
parameter space: this was first shown also to be a sufficient condition
in the case that $P$ and $Q$ are {\em postcritically finite} \cite{tl},
and subsequently for more general classes of $P$ and $Q$.

It is also possible to mate certain Kleinian groups with
polynomial maps. To realise such matings we have to move into the
larger world of {\em holomorphic correspondences}. A {\it
holomorphic correspondence}, of bidegree $m:n$, on the Riemann
sphere $\hat{\mathbb C}$, is a multivalued map $z \to w$ defined
by a relation $p(z,w)=0$, where $p$ is a polynomial of degree $m$
in $z$ and $n$ in $w$. We require that $p(z,w)$ has no square
factors, so that a generic point $w$ has $m$ inverse images $z$
and a generic point $z$ has $n$ images $w$. Equivalently, a
holomorphic correspondence on $\hat{\mathbb C}$ is defined by a
(singular) Riemann surface in $\hat{\mathbb C} \times \hat{\mathbb
C}$ with the two projections to $\hat{\mathbb C}$
branched-coverings of degrees $m$ and $n$ respectively. Examples
of holomorphic correspondences are rational maps (defined by
$P(z)-w=0$), their inverses (defined by $P(w)-z=0$), and finitely
generated Kleinian groups (defined by $(w-A_1z)(w-A_2z)\cdots
(w-A_nz)=0$, where $A_1,\dots, A_n$ are M\"obius transformations
generating the group in question). We formulate below (in Section
3) what it means to say that a holomorphic correspondence is a
{\em mating} of a particular Kleinian group and a particular
polynomial map. The first examples were described in \cite{bpmat}
and more general constructions were presented in \cite{bcomb}.
These examples and constructions pick out particular classes of
polynomial relations $p(z,w)=0$ and then in appropriate
circumstances identify the resulting correspondences as matings.
Below we show how it is possible to create a mating of a quadratic
map and a representation of the group $C_2*C_3$ to order, by
fitting the pieces together using quasiconformal surgery
\cite{dhpoly}. The key observation that enables us to get
started (Section 4.1 below) is that a certain `pair of pants'
domain associated to a representation of $C_2*C_3$ double covers
an annulus carrying precisely the same combinatorial data as does
a `fundamental annulus' for a quadratic-like map.

\section{The ingredients}

\subsection{The quadratic map}

Given any quadratic map $q:z \to z^2+c$, there is a holomorphic
conjugacy from $z \to z^2$ to $q$ on a neighbourhood of $\infty$,
fixing the point $\infty$ and tangent to the identity map there
\cite{dh}. An {\it equipotential} for $q$ is the image of a circle
$\{Re^{2\pi it}: 0 \le t <1\}$ under this conjugacy. It is a
smooth Jordan curve parametrised by {\it external angle} $t$. The
region bounded by such an equipotential is a simply-connected
domain $V$, mapped $2:1$ by $q$ onto a larger domain $U \supset V$
which also has boundary an equipotential parametrised by external
angle (the restriction $q:V \to U$ is an example of a {\it
quadratic-like map} in the sense of Douady and Hubbard
\cite{dhpoly}). We shall denote the annulus $U-V$ by $A$ and its
inner and outer boundaries by $\partial_1 A$ and $\partial_2 A$
respectively (Figure 1).
\begin{figure}
\includegraphics[0in,0in][3.1in,1.9in]{era76el-fig-1}
\caption{An annulus $A$ for the quadratic map $q:z \to z^2+c$.}
\end{figure}
The map $q$ sends $\partial_1 A$ two-to-one onto $\partial_2 A$.
For future reference we note the existence of an involution $i:z
\to -z$ on $V$ sending each $z \in V$ to the other point which has
the same image in $U$ under $q$, and an involution $j$ on
$\partial_2 A$ given by $t \to 1-t$ on external angles (in fact in
what follows $j$ may be taken to be any {\it smooth
orientation-reversing involution} on $\partial_2 A$).

For simplicity, until the final section of this article we shall
assume that the filled Julia set $K(q)$ is connected. The
corresponding set ${\mathcal M}$ of values of the parameter $c$ is
known as the {\it connectivity locus} or {\it Mandelbrot set}.

\subsection{The Kleinian group}

Up to conjugacy each representation $r$ of $C_2*C_3$ in
$PSL(2,{\mathbb C})$ is determined by a single complex parameter,
the cross-ratio between the fixed points on $\hat{\mathbb C}$ of
the action of the generator $\sigma$ of $C_2$ and those of the
generator $\rho$ of $C_3$. Such a representation comes equipped
with a (unique) involution $\chi$ which exchanges the two fixed
points of $\sigma$ and also those of $\rho$, so that
$\chi\sigma=\sigma\chi$ and $\chi\rho=\rho^{-1}\chi$ \cite{bpkl}.
On the Poincar\'e $3$-disc $\chi$ is simply rotation through $\pi$
around the common perpendicular to the axes of $\sigma$ and
$\rho$. Write $G$ for the group $\langle\sigma,\rho,\chi\rangle$.

The faithful discrete actions $r$ with connected regular set
$\Omega(G)$ form a single quasiconformal conjugacy class, the
class of representations for which one can find simply-connected
fundamental domains for $\sigma$ and $\rho$ with interiors
together covering the whole Riemann sphere (i.e. the conditions of
the simplest form of the Klein Combination Theorem are satisfied) 
\cite{wjh,mas}.
Such fundamental domains may be constructed as illustrated in
Figure 2.
\begin{figure}
\includegraphics[0in,0in][1.5in,2.5in]{era76el-fig-2}
\caption{A fundamental domain $\Delta$ for the group
$G=\langle\sigma,\rho,\chi\rangle$.}
\end{figure}
Here $P$ and $P'$ are the fixed points of $\rho$, $Q$ and $Q'$ are
the fixed points of $\sigma$, $R$ is a fixed point of (the
involution) $\chi\rho$ and $S$ is a fixed point of $\chi\sigma$.
The lines $l,m$ and $n$, joining $R$ to $S$, $Q$ to $S$ and $R$ to
$P$, are chosen such that they are smooth and remain
non-intersecting in the quotient orbifold $\Omega(G)/G$. The
region bounded by $n,\rho n, \chi n$ and $\chi\rho n$ is a
fundamental domain for $\rho$, and the region exterior to the loop
made up of $m,\sigma m,\chi m$ and $\chi\sigma m$ is a fundamental
domain for $\sigma$. The intersection of these two regions is a
fundamental domain for the (faithful) action of $C_2*C_3$ on
$\Omega(G)$, and the half $\Delta$ of this intersection bounded by
$n,l,m,\sigma m, \chi l$ and $\rho n$ is a fundamental domain for
the action of $G$. The union of all translates of $\Delta$ under
elements of $C_2*C_3$ is a topological disc $D$ which is a
fundamental domain for the action of $\chi$ on $\Omega(G)$. The
complement $\Lambda(G)$ of $\Omega(G)= D \cup \chi(D)$ in
$\hat{\mathbb C}$ is a Cantor set.

\section{The definition of a mating and the statement of the Theorem}

We say that a $2:2$ holomorphic correspondence $f:\hat{\mathbb C}
\to \hat{\mathbb C}$ is a {\it mating} of the quadratic map $q:z
\to z^2+c$ (where $c\in {\mathcal M}$) and the faithful discrete
representation $r$ of $C_2*C_3$ (having connected regular set) if
$\hat{\mathbb C}$ is partitioned into an open set $\Omega$ and a
closed set $\Lambda$, each completely invariant under $f$ and
with the following properties:

\begin{list}{(\Roman{bean1})}{\usecounter{bean1}\setlength{\rightmargin}{\leftmargin}}
\item $\Lambda$ is the disjoint union of two sets $\Lambda_+$ and
$\Lambda_-$, on which the $2:2$ correspondence $f:\Lambda \to
\Lambda$ decomposes into the following parts:

\begin{list}{(\roman{bean2})}{\usecounter{bean2}\setlength{\rightmargin}{\leftmargin}}
\item $f:\Lambda_- \to \Lambda_-$, a $2:1$ correspondence (a map of
degree two);

\item $f:\Lambda_+ \to \Lambda_+$, a $1:2$ correspondence (the
inverse of a map of degree two);

\item $f:\Lambda_- \to \Lambda_+$, a $1:1$ correspondence (a
bijection).
\end{list}

\item There is a homeomorphism from the filled Julia set
$K$ of $q$ to $\Lambda_-$ conjugating $q\vert_K$ to
$f\vert_{\Lambda_-}$ and a homeomorphism from $K$ to
$\Lambda_+$ conjugating $q\vert_K$ to $f^{-1}\vert_{\Lambda_+}$.
Both are conformal on the interior of $K$.
\item The $2:2$ correspondence $f:\Omega \to \Omega$ acts properly
discontinuously and there is a conformal homeomorphism $h$ from
the orbifold $\Omega/f$ to the orbifold $\Omega(G)/G$ compatible
with the actions of $f$ and $G$ respectively, in the following
sense: there exist a (completely invariant) set of curves
${\mathcal C}$ in $\Omega$ and a fundamental domain $D$ for the
action of $\chi$ on $\Omega(G)$ which is invariant under
$C_2*C_3$, such that $h$ lifts to a conformal homeomorphism
$(\Omega-{\mathcal C}) \to D$ conjugating $f$ to
$\{\sigma\rho,\sigma\rho^{-1}\}$.
\end{list}

\begin{theorem}For every quadratic map $q:z \to z^2 + c$ with
$c \in {\mathcal M}$ and every faithful discrete representation
$r$ of $C_2*C_3$ in $PSL(2,{\mathbb C})$ having connected regular
set, there exists a polynomial relation $p(z,w)=0$ defining a
$2:2$ correspondence $z \to w$ which is a mating of $q$ with $r$.
\end{theorem}

A modification dealing with the case when $c$ is outside
${\mathcal M}$ will be outlined in the final section of this
paper.

\section{The construction and the proof of the Theorem}

\subsection{An annulus associated to the Kleinian group, and
a $2:2$ correspondence on it}

The quotient orbifold $\Omega(G)/G$ is the Riemann surface (with
cone point singularities) obtained from $\Delta$ by making the
boundary identifications corresponding to $\rho,\sigma$ and
$\chi$. Covering this orbifold we have an annulus $B$ consisting
of three contiguous copies of $\Delta$ in $\Omega(G)$, namely
$\Delta \cup \rho\Delta \cup \rho^{-1}\Delta$, with the boundary
identifications (induced by $\chi$) indicated in Figure 3.
\begin{figure}
\includegraphics[0in,0in][4.9in,1.5in]{era76el-fig-3}
\caption{The set $(\Delta \cup \rho\Delta \cup \rho^{-1}\Delta)$
and its quotient, the annulus $B$.}
\end{figure}
Concretely we may regard $B$ as a subset of the quotient Riemann
sphere $\hat{\mathbb C}/\chi$.
We remark that $\Delta \cup \rho\Delta \cup \rho^{-1}\Delta$ is
itself a fundamental domain for the action of the index three
subgroup $\langle\chi,\sigma,\rho\sigma\rho \rangle$ of $G$, and that $\Delta
\cup \rho\Delta \cup \rho^{-1}\Delta$ and its image under $\chi$
make up a `pair of pants' fundamental domain for
$\langle\sigma,\rho\sigma\rho \rangle$, the annulus $B$ being the quotient of
this `pair of pants' by the action of $\chi$.

The rotation $\rho$ maps $\Delta \cup \rho\Delta \cup
\rho^{-1}\Delta$ to itself in the obvious way, with a single fixed
point at $P$, but since $\rho$ {\em anticommutes} with $\chi$, this
action does not descend to a rotation on the quotient annulus $B$.
Rather, the action of the pair $\{\rho,\rho^{-1}\}$ on $\Delta
\cup \rho\Delta \cup \rho^{-1}\Delta$ descends to the action of a
$2:2$ correspondence $g$ on $B$. Under this $2:2$ correspondence
each $z \in B$ is mapped to the points $\rho z$ and $\rho^{-1}z$
(or rather to their equivalence classes under the action of
$\chi$). The points $P$ and $T$ are singular for $g$, having {\em
unique} images $P$ and $R$, respectively, but all other points of
$B$ have two distinct images under $g$, and also $2$ distinct
inverse images, since $g=g^{-1}$. Note that in a neighbourhood of
$P$ the correspondence $g$ behaves like a pair of rotations
through $2\pi/3$ and $-2\pi/3$, but in a neighbourhood of $T$ it
behaves like a square root map. Generic orbits of $g$ have
cardinality three, the correspondence $g$ sending each point of an
orbit to the other two, and the image of $\Delta$ in $B$ is a
`fundamental domain' for the action. The boundary of $B$ is
divided into three segments (two inner and one outer, Figure 3),
each of which is mapped to the other two by $g$. Thus when its
domain is restricted to the inner boundary $\partial_1 B$, and its
range is restricted to the outer boundary $\partial_2 B$, the
correspondence $g$ defines a two to one map. When restricted to a
correspondence from the inner boundary to itself, $g$ defines a
(fixed point free) bijection. Moreover, since the involution
$\sigma$ commutes with $\chi$, it descends to an involution (which
we shall also denote $\sigma$) on the outer boundary $\partial_2
B$ of $B$, having fixed points $Q$ and $S$. Observe that since the
composition $\sigma \circ g:\partial_1 B \to \partial_2 B$ is an
orientation preserving two to one map, and the bijection
$g:\partial_1 B \to \partial_1 B$ is the covering involution of
this map, the annulus $B$ carries the same data as that furnished
on the annulus $A$ by the quadratic map $q$.

\subsection{A bijection between the annuli $A$ and $B$}

In general the annuli $A$ and $B$ will not be conformally
equivalent: the conformal equivalence class of an annulus is
determined by its {\em modulus}, a positive real number \cite{ahl}.
However,
any two annuli are {\em quasiconformally} equivalent.

\begin{lemma} There exists a quasiconformal homeomorphism
$h$ from $A$ to $B$ which
restricts to a smooth homeomorphism from $\partial A$ to $\partial
B$ conjugating the boundary maps $(q:\partial_1 A \to
\partial_2 A,\ j:\partial_2 A \to \partial_2 A)$ to the
boundary maps $(\sigma \circ g:\partial_1 B \to
\partial_2 B,\ \sigma: \partial_2 B \to \partial_2 B)$.
\end{lemma}

\begin{proof}
We first use the fixed points of $j$ to divide the outer boundary
$\partial_2 A$ of $A$ into two intervals and choose any smooth
homeomorphism $h$ from one of these intervals to the corresponding
half of $\partial_2 B$ (which has end points $Q$ and $S$). Now
extend $h$ to a smooth homeomorphism from the whole of $\partial_2
A$ to the whole of $\partial_2 B$, using the involutions $j$ and
$\sigma$, and then to a smooth homeomorphism from $\partial_1 A$
to $\partial_1 B$ by pulling back via $q$ and $\sigma \circ g$.
This gives a smooth $h:\partial A \to \partial B$ equivariant with
respect to the boundary data. But any smooth homeomorphism of
boundaries of annuli extends to a quasiconformal homeomorphism $h$
of the interiors: this follows at once from the corresponding result
for discs \cite{ahl}, since an annulus can be converted into a disc by
cutting along any smooth path joining the inner boundary to the outer.
\end{proof}

Let $\mu$ denote the {\em complex dilatation} $(\partial h
/\partial {\bar z})/(\partial h /\partial z)$ of $h$. By standard
theory of quasiconformal maps \cite{ahl} $\mu$ is of class
$L^\infty$ (bounded almost everywhere) and $\|\mu\|_\infty <1$.

We shall abuse notation to the extent of denoting by $g$ not only
the $2:2$ correspondence on $B$ defined in Section 4.1, but also the
correspondence $g_A=h\circ g_B \circ h^{-1}$ on $A$ obtained
by transporting $g(=g_B)$ from $B$. Thus 
\begin{equation*}\{q:\partial_1 A \to
\partial_2 A\}=\{j\circ g:\partial_1 A \to \partial_2 A\}
\end{equation*} 
and
this degree two map has covering involution 
\begin{equation*}
\{i:\partial_1 A \to
\partial_1 A\}=\{g:\partial_1 A \to \partial_1 A\}.
\end{equation*}
Moreover the
Beltrami differential $\mu$ on $A$ is preserved by $g=g_A:A \to
A$, in the sense that $g^*\mu=\mu$. Since $g_B^*$ is the identity,
$g_B$ being holomorphic, this follows at once from the fact that
\begin{equation*}
g_A^*\mu=(h^{-1})^*\circ (g_B)^* \circ h^*(\mu).
\end{equation*}

\subsection{Constructing the correspondence at the
combinatorial/topological level}

We first glue together $U$ and a second copy $U'$ of $U$, via the
boundary involution $j$, to obtain a sphere $U \cup U'$, equipped
with an involution, which we also denote $j$, exchanging $U$ with
$U'$ and restricting to the original $j$ on the common boundary.
Inside $U'$ we have a simply-connected subdomain $V'$,
corresponding to $V \subset U$. Let $q'=j\circ q \circ j:V' \to
U'$ denote the quadratic map corresponding to $q:V \to U$ and $A'$
denote the annulus $U'-V'$. To define a $2:2$ correspondence $f$
on $U \cup U'$ we fit together:

$\bullet$ $q:V \to U$ (a $2:1$ correspondence);

$\bullet$ $(q')^{-1}=j\circ q^{-1} \circ j:U' \to V'$ (a $1:2$
correspondence);

$\bullet$ $j\circ i:V \to V'$ (a $1:1$ correspondence), and

$\bullet$ $j\circ g:A \to A'$ (a $2:2$ correspondence),

\noindent where $g:A \to A$ is the $2:2$ correspondence
constructed in Section $4.2$ above. We remark that conjugation by
the involution $j$ sends $f$ to $f^{-1}$. Thus $j$ is a {\it
time-reversing symmetry} of $f$.

Using the boundary data identities of Section $4.2$ it is a
straightforward exercise to check that the restrictions of $f$
defined above fit together to define a continuous $2:2$
correspondence $f$ on the whole Riemann sphere. The next step is
to identify the space of grand orbits of mixed iteration of $f$
and $f^{-1}$ on the complement $\Omega$ of $K(q)\cup K(q')$. Let
$A/\sim$ denote the quotient space obtained from the (closed)
annulus $A$ by applying the equivalence relation $g$ on $A$ and
the equivalence relation $\langle g,j\rangle$ on $\partial A$, and let
$\Delta/\approx$ denote the quotient space obtained from $\Delta$
(Figure 2) by identifying $l$ with $\chi l$, $m$ with $\sigma m$
and $n$ with $\rho n$.


\begin{lemma} The grand orbit space of the correspondence $f$,
acting on $\Omega$ by arbitrary combinations of forward and
backward iteration, is homeomorphic to $A/\sim$ and hence to
$\Delta/\approx=\Omega(G)/G$.
\end{lemma}

\begin{proof} We first observe that if $z \in A$,
then $g(z)(\subset A)$ and $j(z)(\in A')$ lie on the grand orbit
of $z$ under $f$. This is because $g(z) \subset f^{-1}\circ f(z)$
and $j(z) \subset f^{-1} \circ  f \circ f^{-1}(z)$. Now we must
show that the grand orbit of any point in $\Omega$ meets $A$ in
a single $g$-orbit. Clearly for each $z\in V\cap \Omega$ there is a unique
positive integer $n$ such that $q^n(z) \in A$ and for each $z' \in
V'\cap \Omega$ there is a unique positive integer $n$ such that $(q')^n(z')
\in A'$, and hence $j\circ(q')^n(z') \in A$ (the claim of
uniqueness needs qualification if $q^n(z) \in
\partial A$ or $(q')^n(z') \in \partial A'$ but it is not hard to
make the appropriate changes). However, in order to show that
$q^n(z)$ and $j\circ(q')^n(z')$ are the {\em only} points of the
grand orbits of $z$ and $z'$ to lie in $A$, we need to check that
no other points of $A$ can be reached by mixing iterations of $q$
and $q'$ with the other branch, $j\circ i$, of $f$ (which, we
recall, carries $V$ bijectively to $V'$). It will suffice to
verify that for any $z \in V$, 
\begin{equation*}
j\circ (q' \circ (j\circ
i))(z)=q(z).
\end{equation*}
However, $j \circ q' \circ j=q$ and $q \circ i=q$, so
we are done.
\end{proof}


\subsection{Making the correspondence
holomorphic}

Since $\partial U$ is smooth and the boundary involution
$j:\partial U \to \partial U$ is smooth, the complex structure on
$U$ extends to a complex structure on the sphere $U \cup U'$. (It
first descends to a complex structure on the quotient $U/j$ and
then lifts to the double cover $U \cup U'$.)

Consider the Beltrami differential $\mu$ on $A$ provided by the
complex dilatation of the quasiconformal homeomorphism $h:A \to B$
(Lemma 1). We may extend $\mu$ to $q^{-1}(A)$ by setting its value
there to be that of the pull-back $q^*\mu$, and we may extend it
to $A'$ by defining its value there to be that of $j^*\mu$. Indeed
by repeatedly pulling back using $q^*$ and $(q')^*$ we may extend
$\mu$ to $U-K=\bigcup q^{-n}(A)$ and $U'-K'=\bigcup
(q')^{-n}(A')$, where $K$ and $K'$ are the filled Julia sets of
$q$ and $q'$ respectively. Finally by defining it to be zero on
$K\cup K'$ we may extend $\mu$ to an $L^\infty$ Beltrami
differential $\mu$ on the whole of the Riemann sphere. Since
$\|\mu\|<1$, we may now apply the Measurable Riemann Mapping
Theorem \cite{ab,ahl} and deduce that there exists a
quasiconformal homeomorphism $\phi:\hat{\mathbb C} \to
\hat{\mathbb C}$ with complex dilatation $\mu$. But $f^*\mu=\mu$,
since $g^*\mu=\mu, j^*\mu=\mu$ and $q^*\mu=\mu$ on the appropriate
regions. Thus, by the chain rule, $\phi \circ f \circ \phi^{-1}$
is holomorphic, except possibly at branch points. But the latter
are removable and hence the $2:2$ correspondence $\phi \circ f
\circ \phi^{-1}$ is holomorphic everywhere. Since $\mu$ vanishes
on $K \cup K'$, and is the complex dilatation of $h$ on $A$, the
correspondence is a mating (in the sense of the definition in
Section 3) of the quadratic map $q$ and the representation $r$
used in its construction, with the union of the images of $\Delta$
under $C_2*C_3$ as the fundamental domain $D$ for the action of
$\chi$ on $\Omega(G)$, and with the grand orbit under $f$ of the
image (under $h^{-1}$) in $A$ of the curve $l \subset \partial
\Delta$ as the set of curves ${\mathcal C}$ such that $\Omega -
{\mathcal C}$ is conformally homeomorphic to $D$.
\begin{figure}
\includegraphics[0in,0in][3.9in,2.1in]{era76el-fig-4}
\caption{Pre-images of the annuli $A$ and $A'$, and cut lines
${\mathcal C}$.}
\end{figure}
In Figure 4, where the coordinates have been chosen so
that $j$ is the map $z \to -z$, we illustrate the annuli $A$ and
$A'$, their first few pre-images under $q^{-1}$ and $(q')^{-1}$
respectively, and the intersection of ${\mathcal C}$ with these
annuli and pre-images.
Note that each $q^{-n}(A)$ is an annulus,
regularly $2^n$-fold covering $A$ itself, and that the union of
all the annuli $q^{-n}(A)$ and $(q')^{-n}(A')$, when cut along the
set of curves ${\mathcal C}$, opens out to form a disc (containing
the point $\infty$).

With the complex structure defined above, the correspondence $f$
has graph an analytic subvariety $\mathcal S$ of $\hat{\mathbb C}
\times \hat{\mathbb C}$. Such a subvariety is algebraic, by Chow's
Theorem \cite{ch,gaga}, and therefore defined by a polynomial
relation $p(z,w)=0$, quadratic in each of $z$ and $w$ since $f$ is
a $2:2$ correspondence. This completes the proof of Theorem 1.
Moreover since the projection $(z,w) \to z$ of $\mathcal S$ to
$\hat{\mathbb C}$ is a double cover with one double point, over
the fixed point $P$ of $\rho$, and two branch points, over $T$
and the critical value of $q$, it follows by a calculation of
Euler characteristic that ${\mathcal S}$ is of genus zero and
hence, from the analysis in \cite{bpmat}, that following a change
in variable the relation $p(z,w)=0$ can be put in the form
\begin{equation*}
\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)
\left(\frac{aw-1}{w-1}\right)+\left(\frac{aw-1}{w-1}\right)^2=3k
\tag{1}
\end{equation*}
for some value of the (complex) parameters $a$ and
$k$. When the correspondence is taken in this form, the
(time-reversing) involution $j$ mapping the complementary subsets
$U$ and $U'$ of the complex plane bijectively to one another is $z
\to -z$ (as in Figure 4).

\begin{figure}
\includegraphics[0in,0in][4.9in,5.3in]{era76el-fig-5}
\caption{Orbits of correspondence $(1)$ (and zoom around cusp on
right), when $a=4.38+0.09i$ and $k=0.91+0.04i$.}
\end{figure}

In Figure 5 we display a computer plot of orbits of a
correspondence $f$ in the family $(1)$, with the values of the
parameters $a$ and $k$ chosen such that the correspondence is one
of the matings described in the theorem: indeed in this example the
quadratic map is $z \to z^2$.
The figure illustrates the grand orbits of the curves
$n,l,m,\sigma m, \chi l, \rho n$ which make up the boundary of
$\Delta$ in Figure 2, plotted to a certain depth, and a single
grand orbit on $\partial K \cup
\partial K'$, plotted to a greater depth.

In \cite{bpmat} it was observed that {\it all} quadratic maps with
connected Julia sets could be realised in the family of
correspondences $(1)$. The advantage of the present analysis is
that the surgery approach shows that matings of {\it all}
quadratic polynomials having connected Julia sets with {\it all}
faithful discrete representations of $C_2*C_3$ having connected
regular sets are realised in this family. We remark that computer
experiment suggests we can go further: densely in the boundary of
the space of representations of $C_2*C_3$ with connected regular
set $\Omega$ lie the circle-packing representations, each still
discrete and faithful but now having $\Omega$ a disjoint union of
(round) discs. Each such representation is obtained by contracting
an appropriate closed geodesic on the orbifold $\Omega/G$ to a
point, and is characterised by a (rational) rotation number $\nu$
specifying the geodesic. Computer experiment strongly suggests
that within the family $(1)$ we can find a mating of each of these
circle-packing representations with any quadratic map $z \to
z^2+c$ such that $c$ does not lie in the $(1-\nu)$-limb of the
Mandelbrot set, the latter being impossible for elementary
combinatorial reasons. This topic will be explored elsewhere.

Analogous constructions can be made mating representations of
$C_p*C_q$ with polynomial maps of degree $(p-1)(q-1)$ for
arbitrary $p$ and $q$. See \cite{bcomb} for a related method which
applies a generalisation of Klein's Combination Theorem.

\section{The case when the quadratic map has disconnected Julia set}

In the case considered so far, where $q$ has a {\it connected}
Julia set, the construction of the mating is independent of the
choice of equipotential made in order to define the domain $U$
(Section 1.1). When the Julia set is not connected, the critical
value $c$ of $q$ lies in the basin of attraction of $\infty$ and
the choice of equipotential used to define $U$ becomes
significant, since the number $n$ such that $q^n(c) \in A = U-V$
is a topological invariant of the correspondence constructed. Thus
when $c$ lies outside the Mandelbrot set, our initial data need
to include not just the quadratic map $z \to z^2 + c$ but also a
choice of equipotential, which should lie {\it outside} the point
$c$ so that both $U$ and $V$ are simply-connected and $A=U-V$ is
an annulus. We can now construct both a $2:2$ correspondence $f$
and a complex structure respected by it, just as we did in the
case of connected Julia sets. This correspondence is no longer a
mating of the quadratic map $q$ with the representation $r$ in the
strict sense of the definition we gave earlier, since the presence
in $\Omega$ of the critical value $c$ and its pre-images prevent
us from obtaining a conjugacy to an action of $C_2*C_3$ in the way
we did before. Nevertheless there is still a conformal
homeomorphism between the orbit space $\Omega /f$ of the
correspondence and that of the group $G=\langle\sigma,\rho,\chi\rangle$ on its
regular set $\Omega(G)$, so it is clear how to recover the
representation $r$ of $G$ from the correspondence. We also remark
that when the representation $r$ is deformed to one lying on the
boundary of moduli space, by contracting an appropriate geodesic
on the orbit space to a point, certain restrictions come into play
as to what positions are allowed for the critical value $c$. It
seems likely that the effect is to exclude matings of the
circle-packing representation of $C_2*C_3$ having rotation number
$\nu$ with quadratic maps $z \to z^2 +c$ having $c$ lying in the
$(1-\nu)$-wake of ${\mathcal M}$. This question, like that towards
the end of the previous section, will be further explored
elsewhere.


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\end{document}