\magnification=1100 \overfullrule=0pt %%\input zero.tex %%\input mathdefs.tex %%\input symbols.tex %%%%%%%%%% \hsize=15truecm \vsize=20.5truecm \baselineskip=12pt %%%%%%%%%% % Bbb.tex: \newfam\msyfam \font\tenmsy=msbm10 \font\sevenmsy=msbm7 \font\fivemsy=msbm5 \textfont\msyfam=\tenmsy \scriptfont\msyfam=\sevenmsy \scriptscriptfont\msyfam=\fivemsy \def\Bbb{\fam\msyfam\tenmsy} % BLACKBOARD BOLD: \def\A{{\Bbb A}} \def\B{{\Bbb B}} \def\C{{\Bbb C}} \def\D{{\Bbb D}} \def\E{{\Bbb E}} \def\F{{\Bbb F}} \def\G{{\Bbb G}} \def\H{{\Bbb H}} \def\I{{\Bbb I}} \def\J{{\Bbb J}} \def\L{{\Bbb L}} \def\K{{\Bbb K}} \def\M{{\Bbb M}} \def\N{{\Bbb N}} \def\O{{\Bbb O}} \def\P{{\Bbb P}} \def\Q{{\Bbb Q}} \def\R{{\Bbb R}} \def\T{{\Bbb T}} \def\U{{\Bbb U}} \def\V{{\Bbb V}} \def\W{{\Bbb W}} \def\X{{\Bbb X}} \def\Y{{\Bbb Y}} \def\Z{{\Bbb Z}} % CALLIGRAPHIC CHARACTERS: \def\AA{{\cal A}} \def\BB{{\cal B}} \def\CC{{\cal C}} \def\DD{{\cal D}} \def\EE{{\cal E}} \def\FF{{\cal F}} \def\GG{{\cal G}} \def\HH{{\cal H}} \def\II{{\cal I}} \def\JJ{{\cal J}} \def\KK{{\cal K}} \def\LL{{\cal L}} \def\MM{{\cal M}} \def\NN{{\cal N}} \def\OO{{\cal O}} \def\PP{{\cal P}} \def\QQ{{\cal Q}} \def\RR{{\cal R}} \def\SS{{\cal S}} \def\TT{{\cal T}} \def\UU{{\cal U}} \def\VV{{\cal V}} \def\WW{{\cal W}} \def\XX{{\cal X}} \def\YY{{\cal Y}} \def\ZZ{{\cal Z}} %%%%%%%%% \def\dim{\hbox{dim}\,} \def\lset{\{\ } % for { \def\st{\ |\ } % for | \def\rset{\ \}} % for } \def\set#1{\lset #1 \rset} \def\sett#1#2{\lset #1 \st #2 \rset} \def\map{\longrightarrow} \def\commadots{\ ,\ldots ,\ } \def\half{{1\over 2}} \def\del{\partial} \def\section#1 {\goodbreak\bigskip {\noindent\hglue-1.5pt\bf#1} \vglue2pt \par\nobreak\noindent } %%%%%%%%%$ \font\medbold=cmbx10 scaled 1100 \def\Heading#1{ \noindent {\medbold #1 } } \def\heading#1{ \noindent {\bf #1 } } \def\address{ \medskip \begingroup \obeylines \parskip=0pt \parindent=0pt \baselineskip=12pt \bf } \def\endaddress{\endgroup} \def\abstract#1//{ \bigskip \begingroup \baselineskip=11pt \noindent{\bf Abstract.} \noindent{\rm#1} \endgroup } \def\frtitle#1//{ \bigskip \noindent {\bf #1} } \def\frabstract#1//{ \bigskip \begingroup \baselineskip=11pt \noindent {\bf R\'esum\'e.} \noindent {\rm#1} \endgroup } \def\proclaim#1{ \goodbreak \medskip \begingroup \noindent {\bf #1} \sl } \def\endproclaim{ \endgroup } \def\bibliography{ \bigskip \centerline{{\medbold References }} \bigskip \begingroup \parindent1cm \parskip4pt \baselineskip=11pt } \def\bi#1{% \item{[#1]}% } \def\endbibliography{ \endgroup } \def\cite#1{[#1]} \def\eqrefer#1{$(\xref{#1})$} \def\ip#1/#2/{\left( \; #1,\; #2\; \right)\;} \def\hip#1/#2/{h(#1,\, #2 )\;} \def\emph#1/{{\sl #1}} %%%%%%%%%%% \footnote{}{First author partially supported by an NSF postdoctoral fellowship. Second and third authors partially supported by NSF grant DMS 9625463} \heading{G\'eom\'etrie alg\'ebrique/Algebraic geometry} \bigskip \Heading{A Complex Hyperbolic Structure for Moduli of Cubic Surfaces} \bigskip \heading{Daniel Allcock, James A. Carlson, and Domingo Toledo} \address Department of Mathematics University of Utah Salt Lake City, Utah, USA. E-mail: allcock, carlson and toledo at math.utah.edu \endaddress \abstract We show that the moduli space $M$ of marked cubic surfaces is biholomorphic to $(B^4 - \HH)/\Gamma_0$ where $B^4$ is complex hyperbolic four-space, where $\Gamma_0$ is a specific group generated by complex reflections, and where $\HH$ is the union of reflection hyperplanes for $\Gamma_0$. Thus $M$ has complex hyperbolic structure, i.e., an (incomplete) metric of constant holomorphic sectional curvature. // \frtitle Une structure hyperbolique complexe pour les modules des surfaces cubiques// \frabstract Nous montrons que l'\'espace des modules $M$ des surfaces cubiques marqu\'ees est biholomorphe \`a $(B^4 - \HH)/\Gamma_0$ o\'u $B^4$ est l'\'espace complexe hyperbolique de dimenson quatre, o\'u $\Gamma_0$ est un groupe sp\'ecifique gen\'er\'e par des reflections complexes, et o\'u $\HH$ est l'union de l'ensemble d'hyperplans de reflections de $\Gamma_0$. Donc $M$ admet une structure hyperbolique complexe, c'est \`a dire une m\'etrique (incompl\`ete) de courbure holomorphe sectionelle constante. // \bigskip \heading{Version fran\c caise abr\'eg\'ee} A une surface cubique (marqu\'ee) correspond une vari\'et\'e cubique de dimension trois (marqu\'ee), \`a savoir le rev\^etement de $\P^3$ ramifi\'e le long la surface. L'application des periodes $f$ pour ces vari\'et\'es de dimension trois est d\'efinie sur l'\'espace des modules $M$ des cubiques marqu\'ees, et cette application $f$ prend ses valeurs dans une quotient de la boule unitaire dans $\C^4$ par l'action du groupe de monodromie projective. Ce groupe $\Gamma_0$ est gen\'er\'e par des reflections complexes dans un ensemble d'hyperplans dont la r\'eunion nous notons par $\HH$. Alors nous avons le resultat suivant: \proclaim{Th\'eor\`eme.} L'application des periodes d\'efinit une biholomorphisme $$ f: M \map \left(B^4 - \HH \right)/\Gamma_0. $$ \endproclaim \noindent De ce th\'eor\`eme on obtient des resultats sur la structure m\'etrique de $M$ et sur son groupe fondamentale: \proclaim{Corollaires.} (1) L'\'espace $M$ des modules des surfaces cubiques marqu\'ees admet une structure hyperbolique complexe: une m\'etrique (incompl\`ete) de courbure holomorphe sectionelle constante. (2) Le groupe fondamentale de $M$ contient un sousgroupe normale qui n'est pas de g\'eneration finie. (3) Le groupe fondamentale de $M$ n'est pas un r\'eseau dans un groupe de Lie semisimple. \endproclaim \medskip \noindent{\bf R\'emarque.} Nos m\'ethodes montrons aussi que la complet\'ee m\'etrique de $(B^4 - \HH)/\Gamma_0$ est l'orbifolde $B^4/\Gamma_0$, isomorphe \`a l'\'espace des modules des surfaces cubiques marqu\'ees stables. Afin de pr\'eciser la notion de surface cubique lisse marqu\'ee, fixons un r\'eseau $L$, une $\Z$-module libre avec une base $e_0 \commadots e_6$ qui est muni de la forme quadratique telle que la base soit orthogonale et telle que $\ip e_0/e_0/ = 1$, $\ip e_k/e_k/ = -1$ pour $k > 0$. Soit $\eta = 3e_0 - (e_1 + \cdots + e_6)$. Alors une \emph surface cubique marqu\'ee/ est compos\'ee d'une surface cubique lisse $S$ et d'une isom\'etrie $\psi: L \map H^2(S,\Z)$ qui envoie $\eta$ sur la classe d'un hyperplan. L'ensemble $M$ de classes d'isomorphisme des surfaces cubiques marqu\'ees porte la structure d'une vari\'et\'e et de plus est une \'espace de modules fines. Une construction de cette \'espace a et\'e don\'ee dans \cite{9}, o\'u on trouve aussi une compactification lisse $C$ de $M$ telle que les points de $C - M$ forme un diviseur \`a croisements normales. Pour definir le groupe $\Gamma_0$, soit $\EE$ l'anneau des entiers d'Eisenstein $\Z[\omega]$ o\'u $\omega = (-1 + \sqrt{-3})/2$, et consid\'erons le produit Cartesian $\EE^5$ muni d'une forme hermitien $\hip v/w/ = -v_1\bar w_1 + v_2\bar w_2 + v_3\bar w_3 + v_4\bar w_4 + v_5\bar w_5$. Alors $(\EE^5,h)$ est l'unique r\'eseau autoduale sur les entiers d'Eisenstein qui est de signature $(4,1)$. Donc $Aut(\EE^5,h)$ est un sousgroupe discr\`ete du groupe unitaire $U(h)$, qui agit sur $B^4 = \sett{ \ell \in \P^4}{ h|\ell < 0 }$. Notons que $\EE/\sqrt{-3}\EE \cong \F_3$ est un corps de trois el\'ements et notons aussi qu'il y a une homomorphisme naturelle $Aut(\EE^5,h) \map Aut(\F_3^5,q)$ o\'u $q$ est la forme quadratique obtenue par reduction de $h$ modulo $\sqrt{-3}$. Notons par ``$P$'' la projectivization, et d\'efinissons un groupe $\Gamma_0$ d'automorphismes de $B^4$ par la suite exacte $$ 1 \map \Gamma_0 \map PAut(\EE^5,h) \map PAut(\F_3^5,q) \map 1. $$ Ce groupe est le groupe discr\`ete du th\'eor\`eme principale. Les hyperplans de $\HH$ sont d\'efinis par les equations $h(x,v) = 0$ pour des vecteurs $v$ dans $\EE^5$ avec $ h(v) = 1$. Notons aussi que $PAut(\F_3^5,q)$ est isomorphe au groupe de Weyl du r\'eseau $E_6$. \section{1. Main results} To a (marked) cubic surface corresponds a (marked) cubic threefold defined as the triple cover of $\P^3$ ramified along the surface. The period map $f$ for these threefolds is defined on the moduli space $M$ of marked cubic surfaces and takes its values in the quotient of the unit ball in $\C^4$ by the action of the projective monodromy group. This group $\Gamma_0$ is generated by complex reflections in a set of hyperplanes whose union we denote by $\HH$. Then we have the following result: \proclaim{Theorem.} The period map defines a biholomorphism $$f: M \map \left(B^4 - \HH\right)/\Gamma_0.$$ \endproclaim \noindent From this identification we obtain results on the metric structure and the fundamental group: \proclaim{Corollaries.} (1) The moduli space of marked cubic surfaces carries a complex hyperbolic structure: an (incomplete) metric of constant holomorphic sectional curvature. (2) The fundamental group of the space of marked cubic surfaces contains a normal subgroup which is not finitely generated. (3) The fundamental group of the space of marked cubic surfaces is not a lattice in a semisimple Lie group. \endproclaim \medskip \noindent{\bf Remark.} Our methods also show that the metric completion of $(B^4 - \HH)/\Gamma_0$ is the complex hyperbolic orbifold $B^4/\Gamma_0$, which is isomorphic to the moduli space of marked stable cubic surfaces. To make precise the notion of smooth marked cubic surface, fix the lattice $L$ to be the free $\Z$-module with basis $e_0 \commadots e_6$ endowed with the quadratic form for which the given basis is orthogonal and such that $\ip e_0/e_0/ = 1$, $\ip e_k/e_k/ = -1$ for $k > 0$. Let $\eta = 3e_0 - (e_1 + \cdots + e_6)$. Then a \emph marked cubic surface/ consists of a smooth cubic surface $S$ and an isometry $\psi: L \map H^2(S,\Z)$ which carries $\eta$ to the hyperplane class. The set $M$ of isomorphism classes of marked cubic surfaces has the structure of a variety and is a fine moduli space. A construction of it is described in \cite{9}, and a smooth compactification $C$ is given for which the points of $C - M$ constitute a normal crossing divisor. To define the group $\Gamma_0$, let $\EE$ denote the ring of Eisenstein integers $\Z[\omega]$ where $\omega = (-1 + \sqrt{-3})/2$ is a cube root of unity, and consider the Cartesian product $\EE^5$ endowed with the hermitian inner product $\hip v/w/ = -v_1\bar w_1 + v_2\bar w_2 + v_3\bar w_3 + v_4\bar w_4 + v_5\bar w_5$. Then $(\EE^5,h)$ is the unique self-dual lattice over the Eisenstein integers with signature $(4,1)$. Thus $Aut(\EE^5,h)$ is a discrete subgroup of the unitary group $U(h)$, which acts on $B^4 = \sett{ \ell \in \P^4}{ h|\ell < 0 }$. Observe that $\EE/\sqrt{-3}\EE \cong \F_3$ is a field of three elements and that there is a natural homomorphism $Aut(\EE^5,h) \map Aut(\F_3^5,q)$ where $q$ is the quadratic form obtained by reduction of $h$ modulo $\sqrt{-3}$. Let ``$P$'' denote projectivization, and define a group $\Gamma_0$ of automorphisms of $B^4$ by $$ 1 \map \Gamma_0 \map PAut(\EE^5,h) \map PAut(\F_3^5,q) \map 1 $$ This is the discrete group of the main theorem. The hyperplanes of $\HH$ are defined by the equations $h(x,v) = 0$ for vectors $v$ in $\EE^5$ with $ h(v) = 1 $. Note that $PAut(\F_3^5,q)$ is isomorphic to the Weyl group of the $E_6$ lattice. \section{2. Construction of a period mapping} To construct the period mapping, we examine in detail the Hodge structures for the cubic threefolds. The underlying lattice $H^3(T,\Z)$ is ten-dimensional, carries unimodular symplectic form $\Omega$, and admits a Hodge decomposition of the form $H^3(T,\C) = H^{2,1} \oplus H^{1,2}$. Choose a generator $\sigma$ for the group of automorphisms of $T$ over $\P^3$, and note that it operates without fixed points on $H^3(T,\Z)$. This action gives $H^3(T,\Z)$ the structure of a five-dimensional module over the Eisenstein integers. It carries a hermitian form $$ \hip x/y/ = \half( \Omega((\sigma - \sigma^{-1})x,\, y) + (\omega - \omega^{-1}) \Omega( x,\, y) ) $$ which is unimodular and of signature $(4,1)$. Now consider the quotient module $H^3(T,\Z)/(1-\omega)H^3(T,\Z)$ and observe that it can be identified isometrically with $(\F_3^5,q)$. We define a marking of $T$ to be choice of such an isometry, and we claim that a marking of a cubic surface determines a marking of the corresponding threefold. Indeed, if $\gamma$ is a primitive two-dimensional homology class on $S$ the it is the boundary of a three-chain on $T$. Write this as $\del \Gamma = \gamma$ and observe that $\sigma \Gamma$ has the same boundary. Thus the three-chain $c(\gamma) = (1-\sigma)\Gamma$ is a cycle. However, it is well-defined only up to addition of elements $(1-\sigma)\Delta$ where $\Delta$ is a three-cycle on $T$. Thus a homomorphism $$ c: H_2^{prim}(S,\Z) \map H_3(T,\Z)/(1-\sigma) $$ is defined. Since a marking of $S$ can be viewed as a basis of $H_2^{prim}(S,\Z)$, application of $c$ to the basis elements defines a basis of $H_3(T,\Z)/(1-\sigma)$, and this gives the required marking of the threefold. The action of $\sigma$ decomposes $H^3(T,\C)$ into eigenspaces $H^3_\lambda$ where $\lambda$ varies over the primitive cube roots of unity. Because $\sigma$ is holomorphic, the decomposition is compatible with the Hodge decomposition and one has $$ H^3_\omega = H^{2,1}_\omega \oplus H^{1,2}_\omega \qquad H^3_{\bar\omega} = H^{2,1}_{\bar\omega} \oplus H^{1,2}_{\bar\omega} . $$ The dimensions of the Hodge components can be found with the help of Griffiths' Poincar\'e residue calculus \cite{5}. Details for this case are found in \cite{3}, section 5. One finds that $$ \dim H^{2,1}_\omega = 4, \quad \dim H^{1,2}_\omega = 1; \qquad \dim H^{2,1}_{\bar\omega} = 1, \quad \dim H^{1,2}_{\bar\omega} = 4 $$ and from the Hodge-Riemann bilinear relations one finds that $h$ has signature $(4,1)$. Now let $\phi$ be a generator of the one-dimensional space $H^{2,1}_{\bar\omega}$ and let $\gamma_1 \commadots \gamma_5$ be a standard basis of $H^3(T,\Z)$ considered as an $\EE$-module. By this we mean that the $\gamma_k$ are orthogonal and that $h(\gamma_1,\gamma_1) = -1$ and $h(\gamma_k,\gamma_k) = 1$ for $k > 1$. Let $v(\phi,\gamma)$ be the vector in $\C^5$ with components $$ v_k = \int_{\gamma_k} \phi $$ One verifies that $h(v,v) < 0$ where now $h$ is the hermitian form $-|z_1|^2 + |z_2|^2 + \cdots + |z_5|^2$. Thus the line generated by $v(\phi,\gamma)$ defines a point in $B^4 \subset \P^4$, and one checks that $v(\phi,\gamma) \not\in \HH$. By well-known constructions (the work of Griffiths), the period vector defines a holomorphic map from the universal cover of $M$ to the ball which transforms according to the projectivized monodromy representation for marked cubic threefolds. The proof that $\Gamma_0$ is the projective monodromy group relies on the work of Libgober \cite{6} and the first author \cite{1}. Thus our construction yields a period map $f: M \map (B^4 - \HH)/\Gamma_0$. \section{3. Properties of the period mapping} We must now show that $f$ is bijective. For injectivity, consider once again the period vector $v(\phi,\gamma)$. The vectors $\gamma_k$ can be decomposed into eigenvectors $\gamma_k'$ and $\gamma_k''$ for $\sigma$, with eigenvectors $\omega$ and $\bar\omega$, respectively. Let $\hat\gamma_k'$ and $\hat\gamma_k''$ denote elements of the corresponding dual basis. Because $\phi$ is an eigenvector with eigenvalue $\bar\omega$, its integral over $\gamma_k'$ vanishes, so that $$ \phi = \sum_k \hat\gamma_k''\int_{\gamma_k''}\phi = \sum_k \hat\gamma_k''\int_{\gamma_k}\phi $$ Thus the components of $v(\phi,\gamma)$ determine $\phi$ as an element of $H^3_{\bar\omega}$. Consequently the line $\C v(\phi,\gamma)$ determines the complex Hodge structure $H^3_{\bar\omega}$. Viewing the Hodge components of $H^3_{\bar\omega}$ as as subspaces of $H^3(T,\C)$, we may take their conjugates to determine the complex Hodge structure $H^3_\omega$. These two complex Hodge structures determine the Hodge structure on $H^3(T,\Z)$. Thus, by the Torelli theorem of Clemens-Griffiths \cite{4}, the period vector $v(\phi,\gamma)$ determines the cubic threefold $T$ up to isomorphism. It remains to show that $T$, which perforce is a cyclic cubic threefold, determines its ramification locus uniquely. This follows from the fact that the locus in question is a planar component of the Hessian surface. To prove surjectivity we first consider a smooth compactification $C$ of $M$ by a normal crossing divisor $D$, e.g., the one given by Naruki \cite{9}, as well as the Satake compactification $\overline{B^4/\Gamma_0}$, obtained by adding fourty points, the ``cusps,'' each corresponding to a null point of $P(\F_3^5, q)$. By well-known results \cite{2} in complex variable theory, the period map has a holomorphic extension to a map $\bar f$ from $C$ to the Satake compactification. Since $C$ is compact, $\bar f$ is open, and $\overline{B^4/\Gamma_0}$ is connected, we conclude that $\bar f$ is surjective. \section{4. Boundary components} To pass from surjectivity of $\bar f$ to surjectivity of $f$, we must show that $\bar f$ maps the compactifying divisor $D$ to the complement of $(B^4 - \HH)/\Gamma_0)$ in the Satake compactification. To this end write $D$ as a sum of irreducible components, $ D = \bigcup D_i' \cup \bigcup D''_j $, where $D_i'$ parametrizes nodal cubic surfaces via the map to the geometric invariant theory compactification of the moduli space of smooth cubics, and where in the same way the $D_j''$ parametrize cubics with an $A_2$ singularity. Now consider a one-parameter family of cubic surfaces with smooth total space acquiring a node. Its local equation near the node has the form $x^2 + y^2 + z^2 = t$ and the corresponding family of cyclic cubic threefolds has the form $x^2 + y^2 + z^2 + w^3 = t$. The local monodromy of the latter has order six, its eigenvalues are primitive sixth roots of unity, and the space of vanishing cycles is two-dimensional. (These facts are well-known adn the relevant literature and arguments are summarized in \cite{3}, section 6). From \cite{7} we conclude that coefficients of the period vector on vanishing cycles are of the form $ A(t) t^{1/6} + B(t)t^{5/6} $ where $A$ and $B$ are holomorphic. Now the space of vanishing cycles is invariant under the action of $\sigma$ and so constitutes a rank one $\EE$-submodule. One can choose a generator $\delta$ for it so that $h(\delta,\delta) = 1$, and then one has $$ \lim_{t \to 0} \int_\delta \phi = 0 $$ Thus the limiting value of the period vector lies in the orthogonal complement of $\delta$. In other words, $\bar f(D'_i)$ lies in $\HH/\Gamma_0$, as required. Consider next a one-parameter family of cubic surfaces with smooth total space whose central fiber acquires an $A_2$ singularity. Its local equation is $x^2 + y^2 + z^3 = t$ and the corresponding family of cyclic cubic threefolds has local equation $x^2 + y^2 + z^3 + w^3 = t$. In this case the local monodromy is of infinite order. After replacing $t$ by $t^3$ one finds an expansion of the form $ \phi(t) = A(t)(\log t)\,\hat\gamma + \hbox{terms bounded in $t$}$, where $A(0) \ne 0$ and where $\hat\gamma$ is an integer cohomology class which is isotropic for $h$. Consequently the line $\C\phi(t)$ converges to the isotropic line $\C\hat\gamma$ as $t$ converges to zero, hence converges to a cusp in the Satake compactification. \section{6. The corollaries} Finally, we comment on the collaries. Part (a) is immediate. For part (b) let $K$ denote the kernel of the map $\pi_1(M) \map \Gamma_0$. Then $K$ is isomorphic to the fundamental group of $B^4 - \HH$ and it is easy to see that its abelianization is not finitely generated. We remark that $K$ is not free: there are many sets of commuting elements corresponding to normal crossings of $\HH$. For (c) we note first that for lattices in semisimple Lie group of real rank greater than one, the results of Margulis \cite{8} imply finite generation of all normal subgroups. The rank one case can be treated separately, as was shown to us by Michael Kapovich. \bibliography \bi{1} D. Allcock, New complex and quaternion-hyperbolic reflection groups, submitted, http://www.math.utah.edu/\~{}allcock \bi{2} A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem. Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays. J. Differential Geometry 6 (1972), 543--560. \bi{3} J. Carlson and D. Toledo, Discriminant complements and kernels of monodromy representations, 22 pp, submitted, http://www.math.utah.edu/\~{}carlson/eprints.html \bi{4} C. H. Clemens and P. A. Griffiths, the intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972) 281--356. \bi{5} P.A. Griffiths, On the periods of certain rational integrals: I and II, Ann. of Math. {\bf 90} (1969) 460-541 \bi{6} A. Libgober, On the fundamental group of the space of cubic surfaces, Math. Zeit. {\bf 162} (1978) 63--67. \bi{8} B. Malgrange, Int\'egrales asymptotiques et monodromie, Ann. Sci. Ecole Norm. Sup., ser. 4 1974, tome 7, 405--430. \bi{8} G. A. Margulis, Quotient groups of discrete subgroups and measure theory, Funct. Anal. Appl. 12 (1978), 295--305. \bi{9} I. Naruki, Cross ratio variety as a moduli space of cubic surfaces, Proc. London Math. Soc. (3) 45 (1982) 1-30. \endbibliography \end