\delta ).$$ \noindent For $x\not\in A$, we can just let $g(x)=\bbH$. For $x\in A$, we let $g(x)=\bbH\setminus \{ x_n:n\in\bbN\} .$ This is a Borel function, since for any basic open $U\subset\bbC$ we have that $U\subseteq g(x)$ if and only $U\subseteq\bbH$ and no $x_n$ is in $U$. \hfill$\dashv$ \medskip Recall that $S_d$ is the space of discrete subgroups of $PSL_2(\bbR )$. This is a Borel subset of $\f (PSL_2(\bbR ))$ in the Effros Borel structure, and hence a standard Borel space. \medskip \noindent {\bf Lemma 5.3.} {\it There is a Borel map $\varphi :S_d \rightarrow\r$ such that $\bbH /G$ is conformally equivalent to $M_{\varphi (G)}$ for all $G\in S_d$ acting freely on $\bbH$.} \medskip \noindent {\bf Proof.} Note that we can indeed verify in a Borel manner whether $G\in S_d$ acts freely on $\bbH$: This amounts to the claim that for all basic open $U\subseteq\bbH$ we may find a finite sequence $V_0, V_1,\cdots , V_n$ of basic open sets covering $\bar U$ and such that for all $i\leq n$, \medskip (*) for all $g\in G$ with $g\neq 1_G, g\cdot V_i\cap V_i=\emptyset$. \medskip \noindent However (*) is Borel, since it amounts to the assertion that for all $W\subseteq PSL_2(\bbR )$ basic open not containing the identity, if $G\cap W\neq\emptyset$ then there exists $h\in W$ such that $h\cdot V_i\cap V_i=\emptyset$. So let us just fix $G\in S_d$ acting freely on $\bbH $ and describe $\varphi (G)$. First we let $(V_i)$ enumerate the basic open sets which are $G$-discrete, in the sense of meeting each $G$-orbit in at most one point, and have diameter $<1$ in the hyperbolic metric $\rho$ (see 4.B). As in the proof of 3.3, this can be used to give a chart for a representative of $\bbH /G$ in $\r$. The one further problem is in uniformly obtaining a metric. For any $\zeta , \xi\in\bbH$ and $g\in G$ we have $$\text{inf}_{h\in G}\rho (h\cdot\zeta ,\xi )=\text{inf}_{h\in G}\rho (h\cdot\zeta , g\cdot\xi ),$$ \noindent and moreover this quantity is greater than zero if and only if $G\cdot\zeta\neq G\cdot\xi $. In particular $$\text{inf}_{h\in G}\rho (h\cdot\zeta ,\xi )=\text{inf}_{h, g\in G}\rho (h\cdot\zeta , g\cdot\xi ).$$ \noindent Therefore $$\tilde d (G\cdot\zeta , G\cdot\xi )=\text{inf}_{h, g\in G}\rho (h\cdot\zeta , g\cdot\xi )$$ \noindent provides the needed metric on $\bbH /G$. Thus if we let $(a_i)$ enumerate a maximal $G$-discrete subset of $\bbH\cap (\bbQ +i\bbQ )$ (in the sense that $G\cdot a_i\cap G\cdot a_j =\emptyset$ for all $i\neq j)$, $y_{i, i'}= \tilde d (G\cdot a_i, G\cdot a_i'), A_i=\{j:G\cdot a_j\cap V_i\neq \emptyset \}$, and $\zeta_{i,j}$ to be the unique element in $V_i \cap G\cdot a_j$, if it exists, we obtain from $p=((y_{i, i'}), (A_i), (\zeta_{i, i'}))$ an element in $\r^n$ with $M_p$ conformally equivalent to $\bbH /G$. There is the further concern that all these steps can be performed in the Borel context, but this is routine and resembles earlier calculations. \hfill$\dashv$ \medskip Recall that for $x\in\bbR^\bbN$ we let $M(x)$ be the complex manifold $$(\bbD\times\bbC )\setminus \bigcup_{n\in\bbN}\{ 1/(n+2)\}\times \{ x_n+(n+2), x_n+(n+3), x_n+2(n+2)\}$$ \noindent endowed with the inherited complex structure. \medskip \noindent {\bf Lemma 6.3.} {\it There is Borel map $$F:\bbR^\bbN\rightarrow \m^2$$ \noindent such that $M(x)$ and $M_{F(x)}$ are biholomorphic for all $x\in\bbR^\bbN$.} \medskip \noindent {\bf Proof.} This follows the method of 3.3.\hfill$\dashv$ \newpage \begin{center} \section*{References} \end{center} \bigskip Beardon, A.F. [84] {\it A Primer on Riemann Surfaces}, London Math. Soc. Lecture Note Series, {\bf 78}, Cambridge Univ. Press, 1984. Becker, H. and A.S. Kechris [96] {\it The Descriptive Set Theory of Polish Group Actions}, London Math Soc. Lecture Note Series, {\bf 232}, Cambridge Univ. Press, 1996. Becker, J., C.W. Henson and L.A. Rubel [80] First-order Conformal Invariants, {\it Ann. of Math}, {\bf 112} (1980), 123-178. Bedford, T., M. Keane, and C. Series (Eds.) [91] {\it Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces}, Oxford Univ. Press, 1991. Dellacherie, C. and P.-A. Meyer [83] {\it Th\'eorie Discr\`ete du Potentiel}, Hermann, Paris, 1983. Dougherty, R., S. Jackson, and A.S. Kechris [94] The Structure of Hyperfinite Borel Equivalence Relations, {\it Trans. Amer. Math. Soc.}, {\bf 341 (1)} (1994), 193-225. Forster, O. [81] {\it Lectures on Riemann Surfaces}, Springer-Verlag, New York, 1987. Halmos, P.R., J. von Neumann [42] Operator Methods in Classical Mechanics, II, {\it Ann. of Math}., {\bf 43} (1942), 332-350. Harrington, L.A., A.S. Kechris, and A. Louveau [90] A Glimm-Effros Dichotomy for Borel Equivalence Relations, {\it J. Amer. Math. Soc}., {\bf 3} (1990), 903-928. Hjorth, G. [9?] Actions of the Classical Banach Spaces, preprint. Hjorth, G. [9?a] Classification and Orbit Equivalence Relations, preprint. Hjorth, G. and A.S. Kechris [95] Analytic Equivalence Relations and Ulm-type Classifications, {\it J. Symb. Logic}, {\bf 60} (1995), 1273-1300. Hjorth, G., A.S. Kechris, and A. Louveau [98] Borel Equivalence Relations Induced by Actions of the Symmetric Group, {\it Ann. Pure and Appl. Logic}, {\bf 92} (1998), 63-112. . Imayoshi, Y. and M. Taniguchi [92] {\it An Introduction to Teichm\"uller Spaces}, Springer-Verlag, 1992. Jackson, S., A.S. Kechris, and A. Louveau [9?] Countable Borel Equivalence Relations, in preparation. Kaplansky, I. [69] {\it Infinite Abelian Groups}, Revised Ed., Univ. of Michigan Press, Ann Arbor, 1969. Katok, S. [92] {\it Fuchsian Groups}, Univ. of Chicago Press, 1992. Kechris, A.S. [91] Amenable Equivalence Relations and Turing Degrees, {\it J. Symb. Logic} {\bf 56} (1991), 182-194. Kechris, A.S. [92] Countable Sections for Locally Compact Group Actions, {\it Erg. Thy. and Dyn. Syst.}, {\bf 12} (1992), 283-295. Kechris, A.S. [95] {\it Classical Descriptive Set Theory}, Springer-Verlag, 1995. Kechris, A.S. [98] Actions of Polish Groups and Classification Problems, to appear in {\it Analysis and Logic}, Cambridge Univ. Press, 1998. Kodaira, K. [86] {\it Complex Manifolds and Deformation of Complex Structures}, Springer-Verlag, New York, 1986. Patterson, A.T. [88] {\it Amenability}, Math Surveys and Monographs, {\bf 29}, Amer. Math Society. Rudin, W. [66] {\it Real and Complex Analysis}, McGraw-Hill, New York, 1966. Shelah, S. [84] Can You Take Solovay's Inaccessible Away, {\it Israel J. Math}, {\bf 48} (1984), 1-47. Thomas, S., and B. Velickovic [98], On the complexity of the isomorphism relation for finitely generated groups, preprint. Wagon, S. [93] {\it The Banach-Tarski Paradox}, Cambridge Univ. Press, 1993. \vskip 1in \noindent Department of Mathematics, UCLA, Los Angeles, CA 90095; greg@@math.ucla.edu \medskip \noindent Department of Mathematics, Caltech, Pasadena, CA 91125; kechris@@caltech.edu \end{document} --=====================_12443916==_--