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\dateposted{March 28, 2000}
\PII{S 1079-6762(00)00076-7}
\copyrightinfo{2000}{American Mathematical Society}

\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}

\author{W. J. Harvey}
\address{Department of Mathematics, King's College, University of London,
Strand, London WC2R 2LS, United Kingdom}

\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}

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 $p(z,w)=0$.



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).
\caption{An annulus $A$ for the quadratic map $q:z \to z^2+c$.}
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) 
Such fundamental domains may be constructed as illustrated in
Figure 2.
\caption{A fundamental domain $\Delta$ for the group
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:

\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:

\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

\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

\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 $r$.

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

\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.
\caption{The set $(\Delta \cup \rho\Delta \cup \rho^{-1}\Delta)$
and its quotient, the annulus $B$.}
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}.
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 $(\sigma \circ g:\partial_1 B \to
\partial_2 B,\ \sigma: \partial_2 B \to \partial_2 B)$.

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.

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\}
this degree two map has covering involution 
\{i:\partial_1 A \to
\partial_1 A\}=\{g:\partial_1 A \to \partial_1 A\}.
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
g_A^*\mu=(h^{-1})^*\circ (g_B)^* \circ h^*(\mu).

\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 $1:2$

$\bullet$ $j\circ i:V \to V'$ (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

\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 $z \in V$, 
j\circ (q' \circ (j\circ
However, $j \circ q' \circ j=q$ and $q \circ i=q$, so
we are done.

\subsection{Making the correspondence

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 $D$.
\caption{Pre-images of the annuli $A$ and $A'$, and cut lines
${\mathcal C}$.}
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
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).

\caption{Orbits of correspondence $(1)$ (and zoom around cusp on
right), when $a=4.38+0.09i$ and $k=0.91+0.04i$.}

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


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