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%_ ************************************************************************** %_ * The TeX source for AMS journal articles is the publishers TeX code * %_ * which may contain special commands defined for the AMS production * %_ * environment. Therefore, it may not be possible to process these files * %_ * through TeX without errors. To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % 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. \begin{thebibliography}{99} \bibitem{ab} L. Ahlfors and L. Bers, \textit{Riemann's mapping theorem for variable metrics}, Annals of Math. \textbf{72} (1960), 385--404. \MR{22:5813} \bibitem{ahl} L. Ahlfors, \textit{Lectures on Quasiconformal Mappings}, Van Nostrand 1966. \MR{34:336} \bibitem{bpmat} S. Bullett and C. 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