%AMSPPT file with "Geometry And Topology Of Complex Hyperbolic And %CR-Manifolds " \magnification=1200 \input amstex \input psfig \documentstyle{amsppt} \NoBlackBoxes \CenteredTagsOnSplits \hcorrection{.25in} \advance\voffset.25in \advance\vsize-.5in \TagsOnRight %\leftheadtext{Boris Apanasov} \rightheadtext{Complex Hyperbolic and CR-manifolds} \define\Ga{\Gamma} \define\ga{\gamma} \define\a{\alpha} \define\da{\delta} \define\Da{\Delta} \define\e{\epsilon} \define\La{\Lambda} \define\la{\lambda} \define\sa{\sigma} \define\Sa{\Sigma} \define\Om{\Omega} \define\om{\omega} \define\z{\zeta} \define\vp{\varphi} \define\vP{\varPhi} \define\gtG{{\frak G}} \define\gth{{\frak h}} \define\gtg{{\frak g}} \define\gtS{{\frak S}} \define\gts{{\frak s}} \define\gu{\frak u} \define\gt{\frak t} \define\sca{{\Cal A}} \define\scg{{\Cal G}} \define\sch{{\Cal H}} \define\scc{{\Cal C}} \define\scd{{\Cal D}} \define\sci{{\Cal I}} \define\scj{{\Cal J}} \define\sco{{\Cal O}} \define\scp{{\Cal P}} \define\scs{{\Cal S}} \define\scr{{\Cal R}} \define\sct{{\Cal T}} \define\ba{{\Bbb A}} \define\bb{{\Bbb B}} \define\bc{{\Bbb C}} \define\bg{{\Bbb G}} \define\bh{{\Bbb H}} \define\bi{{\Bbb I}} \define\bk{{\Bbb K}} \define\bn{{\Bbb N}} \define\bo{{\Bbb O}} \define\bp{{\Bbb P}} \define\bq{{\Bbb Q}} \define\br{{\Bbb R}} \define\bS{{\Bbb S}} \define\bz{{\Bbb Z}} \define\ad{\operatorname{Ad}} \define\ch{\bh_{\bc}^} \define\rh{\bh_{\br}^} \define\cb{\bb_{\bc}^n} \define\aut{\operatorname{Aut}} \define\so{\operatorname{SO}^{\circ}} \define\id{\operatorname{id}} \define\im{\operatorname{Im}} \define\re{\operatorname{Re}} \define\is{\operatorname{Isom}} \define\fix{\operatorname{fix}} \define\rank{\operatorname{rank}} \define\tn{\operatorname{thin}_{\e}} \define\tk{\operatorname{thick}_{\e}} \define\mo{\operatorname{mod}} \define\mob{\operatorname{M\ddot ob}} \define\Hom{\operatorname{Hom}} \define\Vol{\operatorname{Vol}} \define\ord{\operatorname{ord}} \define\rint{\operatorname{int}} \define\rext{\operatorname{ext}} \define\ins{\operatorname{int}} \define\arccsch{\operatorname{arccsch}} \define\csch{\operatorname{csch}} \redefine\cos{cos} \redefine\sin{sin} \define\p{\partial} \define\bs{\backslash} \define\sm{\setminus} \define\ra{\rightarrow} \define\hra{\hookrightarrow} \define\col{\,:\,} \define\wt{\widetilde} \define\cdt{\centerdot} \define\ov{\overline} \topmatter \title Geometry And Topology Of Complex Hyperbolic And CR-Manifolds\endtitle \author Boris Apanasov \endauthor \thanks{Research in MSRI was supported in part by NSF grant DMS-9022140.}\endthanks \address Department of Mathematics, University of Oklahoma, Norman, OK 73019 \endaddress \email Apanasov\@ou.edu \endemail \address Sobolev Inst. of Mathematics, Russian Acad. Sci., Novosibirsk, Russia 630090 \hfill \hfill \hfill \endaddress \address Mathematical Sciences Research Institute, Berkeley, CA 94720-5070 \hfill \hfill \hfill \endaddress \keywords Complex hyperbolic geometry, Cauchy-Riemannian manifolds, discrete groups, geometrical finiteness, nilpotent and Heisenberg groups, Bieberbach theorems, fiber bundles, homology cobordisms, quasiconformal maps, structure deformations, Teichm\"uller spaces\endkeywords \subjclass 57, 55, 53, 51, 32, 22, 20 \endsubjclass \abstract\nofrills {ABSTRACT.} We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between K\"ahler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups. %(in the case of the Heisenberg group $\sch_n$). \endabstract \endtopmatter \bigskip \document \head 1. Introduction \endhead This paper presents recent progress in studying topology and geometry of complex hyperbolic manifolds $M$ with {\it variable} negative curvature and spherical Cauchy-Riemannian manifolds with Carnot-Caratheodory structure at infinity $M_{\infty}$. Among negatively curved manifolds, the class of complex hyperbolic manifolds occupies a distinguished niche due to several reasons. First, such manifolds furnish the simplest examples of negatively curved K\"ahler manifolds, and due to their complex analytic nature, a broad spectrum of techniques can contribute to the study. Simultaneously, the infinity of such manifolds, that is the spherical Cauchy-Riemannian manifolds furnish the simplest examples of manifolds with contact structures. Second, such manifolds provide simplest examples of negatively curved manifolds not having {\it constant} sectional curvature, and already obtained results show surprising differences between geometry and topology of such manifolds and corresponding properties of (real hyperbolic) manifolds with constant negative curvature, see \cite{BS, BuM, EMM, Go1, GM, Min, Yu1}. Third, such manifolds occupy a remarkable place among rank-one symmetric spaces in the sense of their deformations: they enjoy the flexibility of low dimensional real hyperbolic manifolds (see \cite{Th, A1, A2} and \S7) as well as the rigidity of quaternionic/octionic hyperbolic manifolds and higher-rank locally symmetric spaces \cite{MG1, Co2, P}. Finally, since its inception, the theory of smooth 4-manifolds has relied upon complex surface theory to provide its basic examples. Nowadays it pays back, and one can study complex analytic 2-manifolds by using Seiberg-Witten invariants, decomposition of 4-manifolds along homology 3-spheres, Floer homology and new (homology) cobordism invariants, see \cite{W, LB, BE, FS, S, A9} and \S5. Complex hyperbolic geometry is the geometry of the unit ball $\cb$ in $\bc^n$ with the K\"ahler structure given by the Bergman metric (compare \cite{CG, Go3}, whose automorphisms are biholomorphic automorphisms of the ball, i.e., elements of $PU(n,1)$. (We notice that complex hyperbolic manifolds with non-elementary fundamental groups are complex hyperbolic in the sense of S.Kobayashi \cite{Kob}.) Here we study topology and geometry of complex hyperbolic manifolds by using spherical Cauchy-Riemannian geometry at their infinity. This CR-geometry is modeled on the one point compactification of the (nilpotent) Heisenberg group, which appears as the sphere at infinity of the complex hyperbolic space $\ch n$. In particular, our study exploits a structural Theorem 3.1 about actions of discrete groups on nilpotent Lie groups (in particular on the Heisenberg group $\sch_n$), which generalizes a Bieberbach theorem for Euclidean spaces \cite{Wo} and strengthens a result by L.Auslander \cite {Au}. Our main assumption on a complex hyperbolic $n$-manifold $M$ is the geometrical finiteness condition on its fundamental group $\pi_1(M)=G\subset PU(n,1)$, which in particular implies that $G$ is finitely generated \cite{Bow} and even finitely presented, see Corollary 4.5. The original definition of a geometrically finite manifold $M$ (due to L.Ahlfors \cite { Ah}) came from an assumption that $M$ may be decomposed into a cell by cutting along a finite number of its totally geodesic hypersurfaces. The notion of geometrical finiteness has been essentially used in the case of real hyperbolic manifolds (of constant sectional curvature), where geometric analysis and ideas of Thurston have provided powerful tools for understanding of their structure, see \cite{BM, MA, Th, A1, A3}. Some of those ideas also work in spaces with pinched negative curvature, see \cite{Bow}. However, geometric methods based on consideration of finite sided fundamental polyhedra cannot be used in spaces of variable curvature, see \S4, and we base our geometric description of geometrically finite complex hyperbolic manifolds on a geometric analysis of their ``thin" ends. This analysis is based on establishing a fiber bundle structure on Heisenberg (in general, non-compact) manifolds which remind Gromov's almost flat (compact) manifolds, see \cite{Gr1, BK}. As an application of our results on geometrical finiteness, we are able to find finite coverings of an arbitrary geometrically finite complex hyperbolic manifold such that their parabolic ends have the simplest possible structure, i.e., ends with either Abelian or 2-step nilpotent holonomy (Theorem 4.9). In another such an application, we study an interplay between topology and K\"ahler geometry of complex hyperbolic $n$-manifolds, and topology and Cauchy-Riemannian geometry of their boundary $(2n-1)$-manifolds at infinity, see our homology cobordism Theorem 5.4. In that respect, the problem of geometrical finiteness is very different in complex dimension two, where it is quite possible that complex surfaces with finitely generated fundamental groups and ``big" ends at infinity are in fact geometrically finite. We also note that such non-compact geometrically finite complex hyperbolic surfaces have infinitely many smooth structures, see \cite{BE}. The homology cobordism Theorem 5.4 is also an attempt to control the boundary components at infinity of complex hyperbolic manifolds. Here the situation is absolutely different from the real hyperbolic one. In fact, due to Kohn--Rossi analytic extension theorem in the compact case \cite{EMM} and to D.Burns theorem in the case when only one boundary component at infinity is compact (see also \cite{NR1, Th.4.4}, \cite{NR2}), the whole boundary at infinity of a complex hyperbolic manifold $M$ of infinite volume is connected (and the manifold itself is geometrically finite if $\dim_{\bc}M\geq 3$) if one of the above compactness conditions holds. However, if boundary components of $M$ are non-compact, the boundary $\p_{\infty} M$ may have arbitrarily many components due to our construction in Theorems 5.2 and 5.3. The results on geometrical finiteness are naturally linked with the Sullivan's stability of discrete representations of $\pi_1(M)$ into $PU(n,1)$, deformations of complex hyperbolic manifolds and Cauchy-Riemannian manifolds at their infinity, and equivariant (quasiconformal or quasisymmetric) homeomorphisms inducing such deformations and isomorphisms of discrete subgroups of $PU(n,1)$. Results in these directions are discussed in the last two sections of the paper. First of all, complex hyperbolic and CR-structures are very interesting due to properties of their deformations, rigidity versus flexibility. Namely, finite volume complex hyperbolic manifolds are rigid due to Mostow's rigidity \cite{Mo1} (for all locally symmetric spaces of rank one). Nevertheless their {\it constant curvature} analogue, real hyperbolic manifolds are flexible in low dimensions and in the sense of quasi-Fuchsian deformations (see our discussion in \S7). Contrasting to such a flexibility, complex hyperbolic manifolds share the super-rigidity of quaternionic/octionic hyperbolic manifolds (see Pansu's \cite{P} and Corlette's \cite{Co1-2} rigidity theorems, analogous to Margulis's \cite{MG1} super-rigidity in higher rank). Namely, due to Goldman's \cite{Go1} local rigidity theorem in dimension $n=2$ and its extension \cite{GM} for $n\geq 3$, every nearby discrete representation $\rho\col G\to PU(n,1)$ of a cocompact lattice $G\subset PU(n-1,1)$ stabilizes a complex totally geodesic subspace $\ch {n-1}$ in $\ch n$, and for $n\geq 3$, this rigidity is even global due to a celebrated Yue's theorem \cite{Yu1}. One of our goals here is to show that, in contrast to that rigidity of complex hyperbolic non-Stein manifolds, complex hyperbolic Stein manifolds are not rigid in general. Such a flexibility has two aspects. Firstly, we point out that the rigidity condition that the group $G\subset PU(n,1)$ preserves a complex totally geodesic hyperspace in $\ch n$ is essential for local rigidity of deformations only for co-compact lattices $G\subset PU(n-1,1)$. This is due to the following our result \cite{ACG}: \proclaim{Theorem 7.1} Let $G\subset PU(1,1)$ be a {\it co-finite} free lattice whose action in $\ch 2$ is generated by four real involutions (with fixed mutually tangent real circles at infinity). Then there is a continuous family $\{f_\a \}$, $-\e<\a<\e$, of $G$-equivariant homeomorphisms in $\ov{\ch 2}$ which induce non-trivial quasi-Fuchsian (discrete faithful) representations $f^*_\a \col G\to PU(2,1)$. Moreover, for each $\a\neq 0$, any $G$-equivariant homeomorphism of $\ov{\ch 2}$ that induces the representation $f^*_\a$ cannot be quasiconformal. \endproclaim This also shows the impossibility to extend the Sullivan's quasiconformal stability theorem \cite{Su2} to that situation, as well as provides the first continuous (topological) deformation of a co-finite Fuchsian group $G\subset PU(1,1)$ into quasi-Fuchsian groups $G_\a=f_\a G f^{-1}_\a\subset PU(2,1)$ with the (arbitrarily close to one) Hausdorff dimension $\dim_H\La(G_\a)>1$ of the limit set $\La(G_\a)$, $\a\neq1$, compare \cite{Co1}. Secondly, we point out that the noncompactness condition in our non-rigidity theorem is not essential, either. Namely, complex hyperbolic Stein manifolds homotopy equivalent to their closed totally {\it real} geodesic surfaces are not rigid, too. Namely, in complex dimension $n=2$, we provide a canonical construction of continuous quasi-Fuchsian deformations of complex surfaces fibered over closed Riemannian surfaces, which we call ``complex bendings" along simple close geodesics. This is the first such deformations (moreover, quasiconformally induced ones) of complex analytic fibrations over a compact base: \proclaim{Theorem 7.2} Let $G\subset PO(2,1)\subset PU(2,1)$ be a given (non-elementary) discrete group. Then, for any simple closed geodesic $\a$ in the Riemann 2-surface $S=H^2_{\br}/G$ and a sufficiently small $\eta_0>0$, there is a holomorphic family of $G$-equivariant quasiconformal homeomorphisms $F_{\eta}: \ov{\ch 2} \ra \ov{\ch 2}$, $-\eta_0<\eta<\eta_0$, which defines the bending (quasi-Fuchsian) deformation $\Cal B_\a\col (-\eta_0,\,\eta_0)\ra \Cal R_0(G)$ of the group $G$ along the geodesic $\a$, $\Cal B_\a(\eta)=F^*_{\eta}$. \endproclaim The constructed deformations depend on many parameters described by the Teichm\"uller space $\Cal T(M)$ of isotopy classes of complex hyperbolic structures on $M$, or equivalently by the Teichm\"uller space $\Cal T(G)=\Cal R_0(G)/PU(n,1)$ of conjugacy classes of discrete faithful representations $\rho\in\Cal R_0(G)\subset \Hom (G, PU(n,1))$ of $G=\pi_1(M)$: \proclaim{Corollary 7.3} Let $S_p=\rh 2/G$ be a closed totally real geodesic surface of genus $p>1$ in a given complex hyperbolic surface $M=\ch 2/G$, $G\subset PO(2,1)\subset PU(2,1)$. Then there is an embedding $\pi\circ\Cal B\col B^{3p-3}\hra \Cal T(M)$ of a real $(3p-3)$-ball into the Teichm\"uller space of $M$, defined by bending deformations along disjoint closed geodesics in $M$ and by the projection $\pi\col \Cal D(M)\ra \Cal T(M)=\Cal D(M)/PU(2,1)$ in the development space $\Cal D(M)$. \endproclaim As an application of the constructed deformations, we answer a well known question about cusp groups on the boundary of the Teichm\"uller space $\sct (M)$ of a (Stein) complex hyperbolic surface $M$ fibering over a compact Riemann surface of genus $p>1$ \cite{AG}: \proclaim {Corollary 7.12} Let $G\subset PO(2,1)\subset PU(2,1)$ be a uniform lattice isomorphic to the fundamental group of a closed surface $S_p$ of genus $p\geq 2$. Then there is a continuous deformation $R\col \br^{3p-3}\to \Cal T(G)$ (induced by $G$-equivariant quasiconformal homeomorphisms of $\ch 2$) whose boundary group $G_{\infty}=R(\infty)(G)$ has $(3p-3)$ non-conjugate accidental parabolic subgroups. \endproclaim Naturally, all constructed topological deformations are in particular geometric realizations of the corresponding (type preserving) discrete group isomorphisms, see Problem 6.1. However, as Example 6.7 shows, not all such type preserving isomorphisms are so good. Nevertheless, as the first step in solving the geometrization Problem 6.1, we prove the following geometric realization theorem \cite{A7}: \proclaim{Theorem 6.2} Let $\phi : G\ra H$ be a type preserving isomorphism of two non-ele\-men\-ta\-ry geometrically finite groups $G,H\subset PU(n,1)$. Then there exists a unique equivariant ho\-meo\-mor\-phism $f_{\phi}\col \La(G)\ra \La(H)$ of their limit sets that induces the isomorphism $\phi$. Moreover, if $\La(G)=\p \ch n$, the homeomorphism $f_{\phi}$ is the restriction of a hyperbolic isometry $h\in PU(n,1)$. \endproclaim We note that, in contrast to Tukia \cite{Tu} isomorphism theorem in the real hyperbolic geometry, one might suspect that in general the homeomorphism $f_{\phi}$ has no good metric properties, compare Theorem 7.1. This is still one of open problems in complex hyperbolic geometry (see \S6 for discussions). \head 2. Complex hyperbolic and Heisenberg manifolds \endhead We recall some facts concerning the link between nilpotent geometry of the Heisenberg group, the Cauchy-Riemannian geometry (and contact structure) of its one-point compactification, and the K\"ahler geometry of the complex hyperbolic space (compare \cite{GP1, Go3, KR}). One can realize the complex hyperbolic geometry in the complex projective space, $$ \ch n =\{[z] \in \bc\bp^n\col \langle z,z\rangle<0\,,\,z\in \bc ^{n,1} \}\,, $$ as the set of negative lines in the Hermitian vector space $\bc ^{n,1}$, with Hermitian structure given by the indefinite $(n,1)$-form $\langle z,w\rangle = z_1\overline w_1+\cdots+z_n\overline w_n-z_{n+1}\overline w_{n+1}$. Its boundary $\p\ch n = \{[z]\in \bc\bp ^{n,1}\col \langle z,z\rangle=0\}$ consists of all null lines in $\bc\bp^n$ and is homeomorphic to the (2n-1)-sphere $S^{2n-1}$. The full group $\is\ch n$ of isometries of $\ch n $ is generated by the group of holomorphic automorphisms (= the projective unitary group $PU(n,1)$ defined by the group $U(n,1)$ of unitary automorphisms of $\bc ^{n,1}$) together with the antiholomorphic automorphism of $\ch n $ defined by the $\bc$-antilinear unitary automorphism of $\bc ^{n,1}$ given by complex conjugation $z\mapsto \bar z$. The group $PU(n,1)$ can be embedded in a linear group due to A.Borel \cite{Bor} (cf. \cite{AX1, L.2.1}), hence any finitely generated group $G\subset PU(n,1)$ is residually finite and has a finite index torsion free subgroup. Elements $g\in PU(n,1)$ are of the following three types. If $g$ fixes a point in $\ch n$, it is called {\it elliptic}. If $g$ has exactly one fixed point, and it lies in $\partial \ch n$, $g$ is called {\it parabolic}. If $g$ has exactly two fixed points, and they lie in $\partial \ch n$, $g$ is called {\it loxodromic}. These three types exhaust all the possibilities. There are two common models of complex hyperbolic space $\ch n $ as domains in $\bc^n$, the unit ball $\Bbb B^n_{\bc}$ and the Siegel domain $\frak S_n$. They arise from two affine patches in projective space related to $\ch n $ and its boundary. Namely, embedding $\bc^n$ onto the affine patch of $\bc\bp^{n,1}$ defined by $z_{n+1}\not=0$ (in homogeneous coordinates) as $A:\bc^n\ra \bc\bp^n$,\, $z \mapsto [(z,1)]$, we may identify the unit ball $\Bbb B^n_{\bc}(0,1)\subset\bc^n$ with $\ch n = A(\Bbb B^n_{\bc})$. Here the metric in $\bc^n$ is defined by the standard Hermitian form $\langle \langle\, ,\,\rangle\rangle$, and the induced metric on ${\Bbb B}^n_{\bc}$ is the Bergman metric (with constant holomorphic curvature -1) whose sectional curvature is between -1 and -1/4. The Siegel domain model of $\ch n $ arises from the affine patch complimentary to a projective hyperplane $H_{\infty}$ which is tangent to $\p\ch n $ at a point $\infty\in\p\ch n$. For example, taking that point $\infty$ as $(0',-1,1)$ with $0'\in\bc^{n-1}$ and $H_{\infty}=\{[z]\in \bc\bp^n \col z_n + z_{n+1} = 0\}$, one has the map $\bold S \col \bc^n \ra \bc\bp^n\bs H_{\infty}$ such that $$ \left( \matrix z' \\z_n \endmatrix\right) \longmapsto\left[ \matrix z' \\ \frac {1}{2} - z_n \\ \frac {1}{2} + z_n \endmatrix\right] \quad \text{where} \quad z'=\left( \matrix z_1 \\ \vdots \\ z_{n-1} \endmatrix\right)\in \bc^{n-1}\,. $$ In the obtained affine coordinates, the complex hyperbolic space is identified with the {\it Siegel domain} $$ \frak S_n = \bold S^{-1}(\ch n) = \{z\in \bc^n \col z_n + \overline z_n > \langle \langle z', z'\rangle \rangle \}\,, $$ where the Hermitian form is $\langle \bold S(z), \bold S(w) \rangle =\langle \langle z',w' \rangle \rangle - z_n - \overline w_n $. The automorphism group of this affine model of $\ch n$ is the group of affine transformations of $\bc^n$ preserving $\frak S_n$. Its unipotent radical is the {\it Heisenberg group} $\sch_n$ consisting of all {\it Heisenberg translations} $$ T_{\xi,v}\col (w',w_n)\mapsto \left(w'+\xi, w_n+\langle\langle\xi,w'\rangle\rangle+ \frac{1}{2}(\langle\langle\xi,xi\rangle\rangle -iv)\right)\,, $$ where $w',\xi\in\bc^{n-1}$ and $v\in\br$. In particular $\sch_n$ acts simply transitively on $\p\ch n\bs\{\infty\}$, and one obtains the {\it upper half space model} for complex hyperbolic space $\ch n$ by identifying $ \bc ^{n-1}\times \br\times [0,\infty)$ and $\ov{\ch n}\bs \{\infty\}$ as $$ (\xi,v,u)\longmapsto \left[\matrix \xi \\ \frac {1}{2}(1-\langle\langle \xi,\xi\rangle\rangle -u+iv) \\ \frac {1}{2}(1+\langle\langle \xi,\xi\rangle\rangle +u-iv) \endmatrix \right]\,, $$ where $(\xi,v,u)\in \bc ^{n-1}\times \br\times [0,\infty)$ are the horospherical coordinates of the corresponding point in $\ov{\ch n}\bs \{\infty\}$ (with respect to the point $\infty \in \p \ch n $, see \cite{GP1}). We notice that, under this identification, the horospheres in $\ch n$ centered at $\infty$ are the horizontal slices $H_t=\{(\xi,v,u)\in \bc ^{n-1}\times \br \times \br_+\col u=t\}$, and the geodesics running to $\infty$ are the vertical lines $c_{\xi,v}(t)=(\xi,v,e^{2t})$ passing through points $(\xi,v)\in \bc ^{n-1}\times \br$. Thus we see that, via the geodesic perspective from ${\infty}$, various horospheres correspond as $ H_t\ra H_u$ with $(\xi,v,t)\mapsto (\xi,v,u)$. The ``boundary plane" $\bc ^{n-1}\times \br\times \{0\}=\p\ch n\bs\{\infty\}$ and the horospheres $H_u=\bc ^{n-1}\times \br\times \{u\}$, $0>> V @>i>> V\rtimes \Gamma/K @>\pi>> \Phi @>>>1\,. \endCD $$ Since any extension of a finite group by a simply connected nilpotent Lie group splits, there is a homomorphism $s: \Phi\rightarrow V\rtimes \Gamma/K$ such that $\pi\circ s=id_\Phi$. For each $A\in \Phi$, we fix an element $(f(A),(g(A),A))\in V\rtimes\Ga$ representing $s(A)$. Since $s$ is a homomorphism, we have $$ g(AB)^{-1}f(AB)^{-1}=A\left(g(B)^{-1}f(B)^{-1}\right)g(A)^{-1}f(A)^{-1} \quad \text{for}\quad A,B\in \Phi\,.\tag4.3 $$ Define $h: \Phi\rightarrow N$ by $h(A)=g(A)^{-1}f(A)^{-1}$. Then (2.4) shows that $h$ is a cocycle. Since $\Phi$ is finite and $N$ is a simply connected nilpotent Lie group, $H^1(\Phi,N)=0 $ due to \cite {LR}. Thus there exists $b\in N$ such that $h(A)=A(b^{-1})b$ for all $A\in \Phi$. On the other hand, $\pi((1,(a,A))K)=\pi((f(A),(g(A),A))K)=A$ for any $\gamma=(a,A) \in \Gamma$. It follows that there is $v_0\in V$ such that $ a^{-1}v_0=h(A)$. This and (4.3) imply that $a^{-1}v_0=A(b^{-1})b$, and hence $baA(b^{-1})=bv_0b^{-1}$. Now consider the group $b\Gamma b^{-1}$ which acts on $bVb^{-1}$., %$b\Gamma b^{-1}\times bV b^{-1}\rightarrow bV b^{-1}$. For any $\gamma=(a,A)\in \Gamma$, the action of the element $b\gamma b^{-1}=(baA(b^{-1}),A)$ on $bVb^{-1}$ is as follows: $$ ((baA(b^{-1}),A),v')\rightarrow baA(b^{-1})A(v')(baA(b^{-1}))^{-1}\,. $$ In particular, $baA(b^{-1})\cdot A(bVb^{-1})\cdot (baA(b^{-1}))^{-1}=bVb^{-1}$. Therefore, $A(bVb^{-1})=bVb^{-1}$ because of $baA(b^{-1})=bv_0 b^{-1}\in bVb^{-1}$, and hence $b\gamma b^{-1}$ preserves $bVb^{-1}$. \line{\hfil\hfil\hfil\qed} \enddemo Now we can apply our description of discrete group actions on a nilpotent group (Theorem 3.1) to study the structure of Heisenberg manifolds. Such manifolds are locally modeled on the $(\sch_n, \sch (n))$-geometry and each of them can be represented as the quotient $\sch_n/G$ under a discrete, free isometric action of its fundamental group $G$ on $\sch_n$, i.e., the isometric action of a torsion free discrete subgroup of $\sch(n)=\sch_n\rtimes U(n-1)$. Actually, we establish fiber bundle structures on all noncompact Heisenberg manifolds: \proclaim{Theorem 3.3} Let $\Ga\subset \sch_n\rtimes U(n-1)$ be a torsion-free discrete group acting on the Heisenberg group $\sch_n= \bc ^{n-1}\times \br $ with non-compact quotient. Then the quotient $\sch_n/ \Ga$ has zero Euler characteristic and is a vector bundle over a compact manifold. Furthermore, this compact manifold is finitely covered by a nil-manifold which is either a torus or the total space of a circle bundle over a torus. \endproclaim The proof of this claim (see \cite{AX1}) is based on two facts due to Theorem 3.1. First, that the discrete holonomy group $\Ga\cong\pi_1(M)$ of any noncompact Heisenberg manifold $M=\sch_n/\Ga$, $\Ga\subset \sch(n)$, has a proper $\Ga$-invariant subspace $\sch_\Ga\subset \sch_n$. And second, the compact manifold $\sch_{\Ga}/\Ga$ is finitely covered by $\sch_{\Ga}/\Ga^*$ where $\Ga^*$ acts on $\sch_{\Ga}$ by translations. The structure of the covering manifold $\sch_G/G^*$ is given in the following lemma. \proclaim{Lemma 3.4} Let $V$ be a connected Lie subgroup of the Heisenberg group $\sch_n$ and $G\subset V$ a discrete co-compact subgroup of $V$. Then the manifold $V/G$ is \roster \item a torus if $V$ is Abelian; \item the total space of a torus bundle over a torus if $V$ is not Abelian. \endroster \endproclaim Though noncompact Heisenberg manifolds $M$ are vector bundles $\sch_n/\Ga\rightarrow \sch_\Ga/\Ga$, simple examples show \cite{AX1} that such vector bundles may be non-trivial in general. However, up to finite coverings, they are trivial \cite{AX1}: \proclaim{Theorem 3.5} Let $\Ga\subset \sch_n\rtimes U(n-1)$ be a discrete group and $\sch_\Ga\subset \sch_n$ a connected $\Ga$-invariant Lie subgroup on which $\Ga$ acts co-compactly. Then there exists a finite index subgroup $\Ga_0\subset \Ga$ such that the vector bundle $\sch_n/\Ga_0\rightarrow \sch_\Ga/\Ga_0$ is trivial. In particular, any Heisenberg orbifold $\sch_n/\Ga$ is finitely covered by the product of a compact nil-manifold $\sch_\Ga/\Ga_0$ and an Euclidean space. \endproclaim We remark that in the case when $\Ga\subset \sch_n\rtimes U(n-1)$ is a lattice, that is the quotient $\sch_n/\Ga$ is compact, the existence of such finite cover of $\sch_n/\Ga$ by a closed nilpotent manifold $\sch_n/\Ga_0$ is due to Gromov \cite {Gr} and Buser-Karcher \cite{BK} results for almost flat manifolds. Our proof of Theorem 3.5 has the following scheme. Firstly, passing to a finite index subgroup, we may assume that the group $\Ga$ is torsion-free. After that, we shall find a finite index subgroup $\Ga_0\subset \Ga$ whose rotational part is ``good". Then we shall express the vector bundle $\sch_n/\Ga_0\rightarrow \sch_\Ga/\Ga_0$ as the Whitney sum of a trivial bundle and a fiber product. We finish the proof by using the following criterion about the triviality of fiber products: \proclaim{Lemma 3.6} Let $F\times_H V$ be a fiber product and suppose that the homomorphism $\rho\col H\rightarrow GL(V)$ extends to a homomorphism $\rho\col F\rightarrow GL(V)$. Then $F\times_H V$ is a trivial bundle, $F\times_H V\cong F/H\times V$. \endproclaim \demo{Proof} The isomorphism $F\times_H V\cong F/H\times V$ is given by $[f,v]\rightarrow (Hf,\rho(f)^{-1}(v))$. %\line{\hfil \hfil \qed %} \enddemo \head 4. Geometrical finiteness in complex hyperbolic geometry \endhead Our main assumption on a complex hyperbolic $n$-manifold $M$ is the geometrical finiteness of its fundamental group $\pi_1(M)=G\subset PU(n,1)$, which in particular implies that the discrete group $G$ is finitely generated. Here a subgroup $G\subset PU(n,1)$ is called {\it discrete} if it is a discrete subset of $PU(n,1)$. The {\it limit set} $\La(G)\subset \p \ch n$ of a discrete group $G$ is the set of accumulation points of (any) orbit $G(y),\, y\in \ch n$. The complement of $\La(G)$ in $\p \ch n$ is called the {\it discontinuity set} $\Om(G)$. A discrete group $G$ is called {\it elementary} if its limit set $\La(G)$ consists of at most two points. An infinite discrete group $G$ is called {\it parabolic} if it has exactly one fixed point $\fix (G)$; then $\La(G)=\fix (G)$, and $G$ consists of either parabolic or elliptic elements. As it was observed by many authors (cf. \cite {MaG}), parabolicity in the variable curvature case is not as easy a condition to deal with as it is in the constant curvature space. However the results of \S 2 simplify the situation, especially for geometrically finite groups. Geometrical finiteness has been essentially used for real hyperbolic manifolds, where geometric analysis and ideas of Thurston provided powerful tools for understanding of their structure. Due to the absence of totally geodesic hypersurfaces in a space of variable negative curvature, we cannot use the original definition of geometrical finiteness which came from an assumption that the corresponding real hyperbolic manifold $M=\bh^n/G$ may be decomposed into a cell by cutting along a finite number of its totally geodesic hypersurfaces, that is the group $G$ should possess a finite-sided fundamental polyhedron, see \cite{Ah}. However, we can define {\it geometrically finite} groups $G\subset PU(n,1)$ as those ones whose limit sets $\La(G)$ consist of only conical limit points and parabolic (cusp) points $p$ with compact quotients $(\La(G)\bs\{p\})/G_p$ with respect to parabolic stabilizers $G_p\subset G$ of $p$, see \cite {BM, Bow}. There are other definitions of geometrical finiteness in terms of ends and the minimal convex retract of the noncompact manifold $M$, which work well not only in the real hyperbolic spaces $\bh^n$ (see \cite {Mar, Th, A1, A3}) but also in spaces with variable pinched negative curvature \cite {Bow}. Our study of geometrical finiteness in complex hyperbolic geometry is based on analysis of geometry and topology of thin (parabolic) ends of corresponding manifolds and parabolic cusps of discrete isometry groups $G\subset PU(n,1)$. Namely, suppose a point $p\in \p \ch n$ is fixed by some parabolic element of a given discrete group $G\subset PU(n,1)$, and $G_p$ is the stabilizer of $p$ in $G$. Conjugating $G$ by an element $h_p\in PU(n,1)\,, h_p(p)=\infty$, we may assume that the stabilizer $G_p$ is a subgroup $G_{\infty}\subset \sch(n)$. In particular, if $p$ is the origin $0\in\sch_n$, the transformation $h_0$ can be taken as the Heisenberg inversion $\sci$ in the hyperchain $\p\ch {n-1}$. It preserves the unit Heisenberg sphere $S_c(0,1)=\{(\xi,v)\in\sch_n\col ||(\xi,v)||_c=1\}$ and acts in $\sch_n$ as follows: $$ \sci(\xi,v)=\left (\frac {\xi}{|\xi|^2-iv}\,,\, \frac {-v}{v^2+|\xi|^4} \right )\quad \text {where}\,\,(\xi,v)\in \sch_n=\bc^{n-1}\times \br \,. \tag4.1 $$ For any other point $p$, we may take $h_p$ as the Heisenberg inversion $\sci_p$ which preserves the unit Heisenberg sphere $S_c(p,1)=\{(\xi,v)\col \rho_c(p,\,(\xi,v))=1\}$ centered at $p$. The inversion $\sci_p$ is conjugate of $\sci$ by the Heisenberg translation $T_p$ and maps $p$ to $\infty$. After such a conjugation, we can apply Theorem 3.1 to the parabolic stabilizer $G_{\infty}\subset \sch(n)$ and get a connected Lie subgroup $\sch_\infty \subseteq \sch_n$ preserved by $G_{\infty}$ (up to changing the origin). So we can make the following definition. \definition{Definition 4.2} A set $U_{p,r}\subset \overline{\ch n}\bs \{p\}$ is called a {\it standard cusp neighborhood of radius} $r>0$ at a parabolic fixed point $p\in \p \ch n$ of a discrete group $G\subset PU(n,1)$ if, for the Heisenberg inversion $\sci_p\in PU(n,1)$ with respect to the unit sphere $S_c(p,1)$, $\sci_p(p)=\infty\,,$ the following conditions hold: \roster \item $U_{p,r}=\sci_p^{-1}\left(\{x\in \ch n\cup \sch_n\col \rho_c(x,\sch_\infty)\geq 1/r \}\right)\,;$ \item $U_{p,r}$ is precisely invariant with respect to $G_p\subset G$, that is: $$ \gamma(U_{p,r})=U_{p,r} \quad \text{for} \,\,\;\gamma\in G_p\quad \text {and}\quad g(U_{p,r})\cap U_{p,r}=\emptyset \quad \text {for} \,\,\,g\in G\backslash G_p\,. $$ \endroster A parabolic point $p\in \p \ch n$ of $G\subset PU(n,1)$ is called a {\it cusp point} if it has a cusp neighborhood $U_{p,r}$. \enddefinition We remark that some parabolic points of a discrete group $G\subset PU(n,1)$ may not be cusp points, see examples in \S 5.4 of \cite{AX1}. Applying Theorem 3.1 and \cite{Bow}, we have: \proclaim {Lemma 4.3} Let $p\in \p \ch n$ be a parabolic fixed point of a discrete subgroup $G$ in $PU(n,1)$. Then $p$ is a cusp point if and only if $(\La(G)\bs \{p\})/G_p$ is compact. \endproclaim This and finiteness results of Bowditch \cite {B} allow us to use another equivalent definitions of geometrical finiteness. In particular it follows that a discrete subgroup $G$ in $PU(n,1)$ is {\it geometrically finite} if and only if its quotient space $M(G)=[\ch n\cup\Om (G)]/G$ has finitely many ends, and each of them is a cusp end, that is an end whose neighborhood can be taken (for an appropriate $r>0$) in the form: $$ U_{p,r}/G_p\approx (S_{p,r}/G_p) \times (0,1]\,, \tag4.4 $$ where $$ S_{p,r}=\p_H U_{p,r}=\sci_p^{-1}\left(\{x\in H_ \bc ^n\cup \sch_n \col \rho_c(x,\sch_\infty)=1/r\}\right)\,. $$ Now we see that a geometrically finite manifold can be decomposed into a compact submanifold and finitely many cusp submanifolds of the form (4.4). Clearly, each of such cusp ends is homotopy equivalent to a Heisenberg $(2n-1)$-manifold and moreover, due to Theorem 3.3, to a compact $k$-manifold, $k\leq 2n-1$. From the last fact, it follows that the fundamental group of a Heisenberg manifold is finitely presented, and we get the following finiteness result: \proclaim{Corollary 4.5} Geometrically finite groups $G\subset PU(n,1)$ are finitely presented. \endproclaim In the case of variable curvature, it is problematic to use geometric methods based on consideration of finite sided fundamental polyhedra, in particular, Dirichlet polyhedra $D_y(G)$ for $G\subset PU(n,1)$ bounded by bisectors in a complicated way, see \cite { Mo2, GP1, FG}. In the case of discrete parabolic groups $G\subset PU(n,1)$, one may expect that the Dirichlet polyhedron $D_y(G)$ centered at a point $y$ lying in a $G$-invariant subspace has finitely many sides. It is true for real hyperbolic spaces \cite{A1} as well as for cyclic and dihedral parabolic groups in complex hyperbolic spaces. Namely, due to \cite{ Ph}, Dirichlet polyhedra $D_y(G)$ are always two sided for any cyclic group $G\subset PU(n,1)$ generated by a Heisenberg translation. Due the main result in \cite{GP1}, this finiteness also holds for a cyclic ellipto-parabolic group or a dihedral parabolic group $G\subset PU(n,1)$ generated by inversions in asymptotic complex hyperplanes in $\ch n$ if the central point $y$ lies in a $G$-invariant vertical line or $\br$-plane (for any other center $y$, $D_y(G)$ has infinitely many sides). Our technique easily implies that this finiteness still holds for generic parabolic cyclic groups \cite{AX1}: \proclaim{Theorem 4.6} For any discrete group $G\subset PU(n,1)$ generated by a parabolic ele\-ment, there exists a point $y_0\in \ch n$ such that the Dirichlet polyhedron $D_{y_0}(G)$ centered at $y_0$ has two sides. \endproclaim \demo{Proof} Conjugating $G$ and due to Theorem 3.1, we may assume that $G$ preserves a one dimensional subspace $\sch_{\infty}\subset \sch_n$ as well as $\sch_{\infty}\times\br_+\subset \ch n$, where $G$ acts by translations. So we can take any point $y_0\in\sch_{\infty}\times\br_+$ as the central point of (two-sided) Dirichlet polyhedron $D_{y_0}(G)$ because its orbit $G(y_0)$ coincides with the orbit $G'(y_0)$ of a cyclic group generated by the Heisenberg translation induced by $G$. \line{\hfil \hfil \qed} \enddemo However, the behavior of Dirichlet polyhedra for parabolic groups $G\subset PU(n,1)$ of rank more than one can be very bad. It is given by our construction \cite{AX1}, where we have evaluated intersections of Dirichlet bisectors with a 2-dimensional slice: \proclaim{Theorem 4.7} Let $G\subset PU(2,1)$ be a discrete parabolic group conjugate to the subgroup $\Ga=\{(m,n)\in \bc\times \br \col m,n\in \bz\}$ of the Heisenberg group $\sch_2=\bc \times \br$. Then any Dirichlet polyhedron $D_y(G)$ centered at any point $y\in \ch 2$ has infinitely many sides. \endproclaim Despite the above example, the below application of Theorem 3.1 provides a construction of fundamental polyhedra $P(G)\subset\ch n$ for arbitrary discrete parabolic groups $G\subset PU(n,1)$, which are bounded by finitely many hypersurfaces (different from Dirichlet bisectors). This result may be seen as a base for extension of Apanasov's construction \cite {A1} of finite sided pseudo-Dirichlet polyhedra in $\bh^n$ to the case of the complex hyperbolic space $\ch n$. \proclaim{Theorem 4.8} For any discrete parabolic group $G\subset PU(n,1)$, there exists a finite-sided fundamental polyhedron $P(G)\subset \ch n$. \endproclaim \demo{Proof} After conjugation, we may assume that $G\subset \sch_n\rtimes U(n-1)$. Let $\sch_{\infty}\subset\sch_n= \bc^{n-1}\times \br$ be the connected $G$-invariant subgroup given by Theorem 3.1. For a fixed $u_0>0$, we consider the horocycle $V_{u_0}=\sch_{\infty}\times \{u_0\}\subset \bc^{n-1}\times \br\times \br_+ =\ch n$. For distinct points $y,y'\in V_{u_0}$, the bisector $C(y,y')=\{z\in\ch n : d(z,y)=d(z,y')\}$ intersects $V_{u_0}$ transversally. Since $V_{u_0}$ is $G$-invariant, its intersection with a Dirichlet polyhedron $$ D_y(G)=\bigcap_{g\in G\backslash \{id\} } \{w\in \ch n \col d(w,y)0$ and $\rho(0)=1$, one can define the length of an edge $[a,b]\subset K(G,\sa)$ as $d_\rho(a,b)=\min \{\rho(|a|), \rho(|b|)\}$. Considering paths of minimal length in the sense of the function $d_\rho(a,b)$, one can extend it to a metric on the Cayley graph $K(G,\sa)$. So taking the Cauchy completion $\ov {K(G,\sa)}$ of that metric space, we have the definition of the group completion $\ov G$ as the compact metric space $\ov {K(G,\sa)}\bs K(G,\sa)$, see \cite{Fl}. Up to a Lipschitz equivalence, this definition does not depend on $\sa$. It is also clear that, for a cyclic group $\bz$, its completion $\ov{\bz}$ consists of two points. Nevertheless, for a nilpotent group $G$ with one end, its completion $\ov G$ is a one-point set \cite{Fl}. Now we can define a proper equivariant embedding $F: K(G,\sa)\hookrightarrow \ch n$ of the Cayley graph of a given geometrically finite group $G\subset PU(n,1)$. To do that we may assume that the stabilizer of a point, say $0\in\ch n$, is trivial. Then we set $F(g)=g(0)$ for any vertex $g\in K(G,\sa)$, and $F$ maps any edge $[a,b]\subset K(G,\sa)$ to the geodesic segment $[a(0),b(0)]\subset \ch n$. \proclaim{Proposition 6.3} For a geometrically finite discrete group $G\subset PU(n,1)$, there are constants $K,K'>0$ such that the following bounds hold for all elements $g\in G $ with $|g|\geq K'$: $$ \ln(2|g|-K)^2 -\ln K^2 \leq d(0,g(0))\leq K|g|\,.\tag6.4 $$ \endproclaim The proof of this claim is based on a comparison of the Bergman metric $d(*,*)$ and the path metric $d_0(*,*)$ on the following subset $bh_0\subset \ch n$. Let $C(\La(G))\subset \ch n$ be the convex hull of the limit set $\La(G)\subset \p\ch n$, that is the minimal convex subset in $\ch n$ whose closure in $\ov{\ch n}$ contains $\La(G)$. Clearly, it is $G$-invariant, and its quotient $C(\La(G))/G$ is the minimal convex retract of $\ch n/G$. Since $G$ is geometrically finite, the complement in $M(G)$ to neighbourhoods of (finitely many) cusp ends is compact and correspond to a compact subset in the minimal convex retract, which can be taken as $\bh_0/G$. In other words, $\bh_0\subset C(\La(G))$ is the complement in the convex hull to a $G$-invariant family of disjoint horoballs each of which is strictly invariant with respect to its (parabolic) stabilizer in $G$, see \cite{AX1, Bow}, cf. also \cite{A1, Th. 6.33}. Now, having co-compact action of the group $G$ on the domain $\bh_0$ whose boundary includes some horospheres, we can reduce our comparison of distances $d=d(x,x')$ and $d_0=d_0(x,x')$ to their comparison on a horosphere. So we can take points $x=(0,0,u)$ and $x'=(\xi,v,u)$ on a ``horizontal" horosphere $S_u=\bc^{n-1}\times\br\times\{u\}\subset \ch n$. Then the distances $d$ and $d_0$ are as follows \cite{Pr2}: $$ \cosh^2 \frac{d}{2} =\frac{1}{4u^2}\left(|\xi|^4+4u|\xi|^2+4u^2+v^2\right)\,, \quad d_0^2=\frac{|\xi|^2}{u} + \frac{v^2}{4u^2}\,.\tag6.5 $$ This comparison and the basic fact due to Cannon \cite{Can} that, for a co-compact action of a group $G$ in a metric space $X$, its Cayley graph can be quasi-isometrically embedded into $X$, finish our proof of (6.4). Now we apply Proposition 6.3 to define a $G$-equivariant extension of the map $F$ from the Cayley graph $K(G,\sa)$ to the group completion $\ov G$. Since the group completion of any parabolic subgroup $G_p\subset G$ is either a point or a two-point set (depending on whether $G_p$ is a finite extension of cyclic or a nilpotent group with one end), we get \proclaim{Theorem 6.6} For a geometrically finite discrete group $G\subset PU(n,1)$, there is a continuous $G$-equivariant map $\Phi_G : \ov G\ra \La(G)$. Moreover, the map $\Phi_G$ is bijective everywhere but the set of parabolic fixed points $p\in\La(G)$ whose stabilizers $G_p\subset G$ have rank one. On this set, the map $\Phi_G$ is two-to-one. \endproclaim Now we can finish our proof of Theorem 6.2 by looking at the following diagram of maps: $$ \CD \La(G) @<{\Phi_G}<< \ov G @>{\ov\phi}>> \ov H @>{\Phi_H}>> \La(H)\,, \endCD $$ where the homeomorphism $\ov\phi$ is induced by the isomorphism $\phi$, and the continuous maps $\Phi_G$ and $\Phi_H$ are defined by Theorem 6.6. Namely, one can define a map $f_\phi=\Phi_H\ov\phi\Phi^{-1}_G$. Here the map $\Phi^{-1}_G$ is the right inverse to $\Phi_G$, which exists due to Theorem 6.6. Furthermore, the map $\Phi^{-1}_G$ is bijective everywhere but the set of parabolic fixed points $p\in\La(G)$ whose stabilizers $G_p\subset G$ have rank one, where it is 2-to-1. Hence the composition map $f_\phi$ is bijective and $G$-equivariant. Its uniqueness follows from its continuity and the fact that the image of the attractive fixed point of an loxodromic element $g\in G$ must be the attractive fixed point of the loxodromic element $\phi(g)\in H$ (such loxodromic fixed points are dense in the limit set, see \cite{A1}). The last claim of the Theorem 6.2 directly follows from the Mostow rigidity theorem \cite{Mo1} because a geometrically finite group $G\subset PU(n,1)$ with $\La(G)=\p \ch n$ is co-finite: $\text{\rm vol}\, (\ch n/G)<\infty$. \line{\hfil \hfil \qed} \enddemo \remark{Remark 6.7} Our proof of Theorem 6.2 can be easily extended to the general situation, that is, to construct equivariant homeomorphisms $f_\phi\col \La(G)\to \La(H)$ conjugating the actions (on the limit sets) of isomorphic geometrically finite groups $G, H\subset \is X$ in a (symmetric) space $X$ with pinched negative curvature $K$, $-b^2\leq K\leq -a^2<0$. Actually, bounds similar to (6.4) in Prop. 6.3 (crucial for our argument) can be obtained from a result due to Heintze and Im Hof \cite{HI, Th.4.6} which compares the geometry of horospheres $S_u\subset X$ with that in the spaces of constant curvature $-a^2$ and $-b^2$, respectively. It gives, that for all $x,y\in S_u$ and their distances $d=d(x,y)$ and $d_u=d_u(x,y)$ in the space $X$ and in the horosphere $S_u$, respectively, one has that $\frac{2}{a}\sinh (a\cdot d/2)\leq d_u \leq \frac{2}{b}\sinh (b\cdot d/2)$. %\frac{2d}{b}\,.$$ \endremark \vskip4pt Upon existence of such homeomorphisms $f_{\varphi}$ inducing given isomorphisms $\varphi$ of discrete subgroups of $PU(n,1)$, the Problem 6.1 can be reduced to the questions whether $f_{\varphi}$ is quasisymmetric with respect to the Carnot-Carath\'eodory (or Cygan) metric, and whether there exists its $G$-equivariant extension to a bigger set (to the sphere at infinity $\p X$ or even to the whole space $\ov{\ch n}$) inducing the isomorphism $\varphi$. For convex cocompact groups obtained by nearby representations, this may be seen as a generalization of D.Sullivan stability theorem \cite{Su2}, see also \cite{A9}. However, in a deep contrast to the real hyperbolic case, here we have an interesting effect related to possible noncompactness of the boundary $\p M(G)=\Om(G)/G$. Namely, even for the simplest case of parabolic cyclic groups $G\cong H \subset PU(n,1)$, the homeomorphic CR-manifolds $\p M(G)=\sch^n/G$ and $\p M(H)=\sch^n/H$ may be not quasiconformally equivalent, see \cite {Min}. In fact, among such Cauchy-Riemannian 3-manifolds (homeomorphic to $\br^2\times S^1$), there are exactly two quasiconformal equivalence classes whose representatives have the holonomy groups generated correspondingly by a vertical $\sch$-translation by $(0, 1)\in \bc\times \br$ and a horizontal $\sch$-translation by $(1, 0)\in \bc\times \br$. Theorem 7.1 presents a more sophisticated topological deformation $\{f_{\a}\}$, $f_{\a}\col \ov{\ch 2}\to\ov{\ch 2}$, of a "complex-Fuchsian" co-finite group $G\subset PU(1,1)\subset PU(2,1)$ to quasi-Fuchsian discrete groups $G_{\a}=f_{\a}Gf_{\a}^{-1}\subset PU(2,1)$. It deforms pure parabolic subgroups in $G$ to subgroups in $G_{\a}$ generated by Heisenberg ``screw translations". As we point out, any such $G$-equivariant conjugations of the groups $G$ and $G_{\a}$ cannot be contactomorphisms because they must map some poli of Dirichlet bisectors to non-poli ones in the image-bisectors; moreover, they cannot be quasiconformal, either. This shows the impossibility of the mentioned extension of Sullivan's stability theorem to the case of groups with rank one cusps. Also we note that, besides the metrical (quasisymmetric) part of the geometrization Problem 6.1, there are some topological obstructions for extensions of equivariant homeomorphisms $f_{\varphi}$, $f_{\varphi}\col \La(G)\ra \La(H)$. It follows from the next example. \proclaim{Example 6.7} Let $G\subset PU(1,1)\subset PU(2,1)$ and $H\subset PO(2,1)\subset PU(2,1)$ be two geometrically finite (loxodromic) groups isomorphic to the fundamental group $\pi_1(S_g)$ of a compact oriented surface $S_g$ of genus $g>1$. Then the equivariant homeomorphism $f_{\varphi}\col \La(G)\ra \La(H)$ cannot be homeomorphically extended to the whole sphere $\p \ch 2 \approx S^3$. \endproclaim \demo{Proof} The obstruction in this example is topological and is due to the fact that the quotient manifolds $M_1=\ch 2/G$ and $M_2=\ch 2/H$ are not homeomorphic. Namely, these complex surfaces are disk bundles over the Riemann surface $S_g$ and have different Toledo invariants: $\tau(\ch 2/G)=2g-2$ and $\tau(\ch 2/H)=0$, see \cite {To}. The complex structures of the complex surfaces $M_1$ and $M_2$ are quite different, too. The first manifold $M_1$ has a natural embedding of the Riemann surface $S_g$ as a holomorphic totally geodesic closed submanifold, and hence $M_1$ cannot be a Stein manifolds. The second manifolds $M_2$ is a Stein manifold due to a result by Burns--Shnider \cite{BS}. Moreover due to Goldman \cite{Go1}, since the surface $S_p\subset M_1$ is closed, the manifold $M_1$ is locally rigid in the sense that every nearby representation $G\rightarrow PU(2,1)$ stabilizes a complex geodesic in $\ch 2$ and is conjugate to a representation $G\rightarrow PU(1,1) \subset PU(2,1)$. In other words, there are no non-trivial ``quasi-Fuchsian" deformations of $G$ and $M_1$. On the other hand, as we show in the next section (cf. Theorem 7.1), the second manifold $M_2$ has plentiful enough Teichm\"uller space of different ``quasi-Fuchsian" complex hyperbolic structures. \line{\hfil \hfil \qed} \enddemo \head 7. Deformations of complex hyperbolic and CR-structures: flexibility versus rigidity \endhead Since any real hyperbolic $n$-manifold can be (totally geodesically) embedded to a complex hyperbolic $n$-manifold $\ch n/G$, flexibility of the latter ones is evident starting with hyperbolic structures on a Riemann surface of genus $g>1$, which form Teichm\"uller space, a complex analytic $(3g-3)$-manifold. Strong rigidity starts in real dimension 3. Namely, due to the Mostow rigidity theorem \cite{M1}, hyperbolic structures of finite volume and (real) dimension at least three are uniquely determined by their topology, and one has no continuous deformations of them. Yet hyperbolic 3-manifolds have plentiful enough infinitesimal deformations and, according to Thurston's hyperbolic Dehn surgery theorem \cite{Th}, noncompact hyperbolic 3-manifolds of finite volume can be approximated by compact hyperbolic 3-manifolds. Also, despite their hyperbolic rigidity, real hyperbolic manifolds $M$ can be deformed as conformal manifolds, or equivalently as higher-dimensional hyperbolic manifolds $M\times (0,1)$ of infinite volume. First such quasi-Fuchsian deformations were given by the author \cite{A2} and, after Thurston's ``Mickey Mouse" example \cite{Th}, they were called bendings of $M$ along its totally geodesic hypersurfaces, see also \cite{A1, A2, A4-A6, JM, Ko, Su1}. Furthermore, all these deformations are quasiconformally equivalent showing a rich supply of quasiconformal $G$-equivariant homeomorphisms in the real hyperbolic space $\rh n$. In particular, the limit set $\La(G)\subset\p\rh {n+1}$ deforms continuously from a round sphere $\p \rh n=S^{n-1}\subset S^n=\rh {n+1}$ into nondifferentiably embedded topological $(n-1)$-spheres quasiconformally equivalent to $S^{n-1}$. Contrasting to the above flexibility, ``non-real" hyperbolic manifolds seem much more rigid. In particular, due to Pansu \cite{P}, quasiconformal maps in the sphere at infinity of quaternionic/octionic hyperbolic spaces are necessarily automorphisms, and thus there cannot be interesting quasiconformal deformations of corresponding structures. Secondly, due to Corlette's rigidity theorem \cite{Co2}, such manifolds are even super-rigid -- analogously to Margulis super-rigidity in higher rank \cite{MG1}. Furthermore, complex hyperbolic manifolds share the above rigidity of quaternionic/octionic hyperbolic manifolds. Namely, due to the Goldman's local rigidity theorem in dimension $n=2$ \cite{G1} and its extension for $n\geq 3$ \cite{GM}, every nearby discrete representation $\rho\col G\to PU(n,1)$ of a cocompact lattice $G\subset PU(n-1,1)$ stabilizes a complex totally geodesic subspace $\ch {n-1}$ in $\ch n$. Thus the limit set $\La(\rho G)\subset \p\ch n$ is always a round sphere $S^{2n-3}$. In higher dimensions $n\geq 3$, this local rigidity of complex hyperbolic $n$-manifolds $M$ homotopy equivalent to their closed complex totally geodesic hypersurfaces is even global due to a recent Yue's rigidity theorem \cite{Yu1}. Our goal here is to show that, in contrast to rigidity of complex hyperbolic (non-Stein) manifolds $M$ from the above class, complex hyperbolic Stein manifolds $M$ are not rigid in general (we suspect that it is true for all Stein manifolds with ``big" ends at infinity). Such a flexibility has two aspects. First, we point out that the condition that the group $G\subset PU(n,1)$ preserves a complex totally geodesic hyperspace in $\ch n$ is essential for local rigidity of deformations only for co-compact lattices $G\subset PU(n-1,1)$. This is due to the following our result \cite{ACG}: \proclaim{Theorem 7.1} Let $G\subset PU(1,1)$ be a {\it co-finite} free lattice whose action in $\ch 2$ is generated by four real involutions (with fixed mutually tangent real circles at infinity). Then there is a continuous family $\{f_\a \}$, $-\e<\a<\e$, of $G$-equivariant homeomorphisms in $\ov{\ch 2}$ which induce non-trivial quasi-Fuchsian (discrete faithful) representations $f^*_\a \col G\to PU(2,1)$. Moreover, for each $\a\neq 0$, any $G$-equivariant homeomorphism of $\ov{\ch 2}$ that induces the representation $f^*_\a$ cannot be quasiconformal. \endproclaim This and an Yue's \cite{Yu2} result on Hausdorff dimension show that there are deformations of a co-finite Fuchsian group $G\subset PU(1,1)$ into quasi-Fuchsian groups $G_\a=f_\a G f^{-1}_\a\subset PU(2,1)$ with Hausdorff dimension of the limit set $\La(G_\a)$ strictly bigger than one. Secondly, we point out that the noncompactness condition in the above nonrigidity is not essential, either. Namely, complex hyperbolic Stein manifolds $M$ homotopy equivalent to their closed totally {\it real} geodesic surfaces are not rigid, too. Namely, we give a canonical construction of continuous non-trivial quasi-Fuchsian deformations of manifolds $M$, $\dim_{\bc}M=2$, fibered over closed Riemann surfaces, which are the first such deformations of fibrations over compact base (for a non-compact base corresponding to an ideal triangle group $G\subset PO(2,1)$, see \cite{GP2}). Our construction is inspired by the approach the author used for bending deformations of real hyperbolic (conformal) manifolds along totally geodesic hypersurfaces (\cite{A2, A4}) and by an example of M.Carneiro--N.Gusevskii \cite {Gu} constructing a non-trivial discrete representation of a surface group into $PU(2,1)$. In the case of complex hyperbolic (and Cauchy-Riemannian) structures, the constructed ``bendings" work however in a different way than in the real case. Namely our complex bending deformations involve simultaneous bending of the base of the fibration of the complex surface $M$ as well as bendings of each of its totally geodesic fibers (see Remark 7.9). Such bending deformations of complex surfaces are associated to their real simple closed geodesics (of real codimension 3), but have nothing common with the so called cone deformations of real hyperbolic 3-manifolds along closed geodesics, see \cite{A6, A9}. Furthermore, there are well known complications in constructing equivariant homeomorphisms in the complex hyperbolic space and in Cauchy-Riemannian geometry, which are due to necessary invariantness of the K\"ahler and contact structures (correspondingly in $\ch n$ and at its infinity, $\ov{\sch_n}$). Despite that, the constructed complex bending deformations are induced by equivariant homeomorphisms of $\ov{\ch n}$, which are in addition quasiconformal: \proclaim{Theorem 7.2} Let $G\subset PO(2,1)\subset PU(2,1)$ be a given (non-elementary) discrete group. Then, for any simple closed geodesic $\a$ in the Riemann 2-surface $S=H^2_{\br}/G$ and a sufficiently small $\eta_0>0$, there is a holomorphic family of $G$-equivariant quasiconformal homeomorphisms $F_{\eta}: \ov{\ch 2} \ra \ov{\ch 2}$, $-\eta_0<\eta<\eta_0$, which defines the bending (quasi-Fuchsian) deformation $\Cal B_\a\col (-\eta_0,\,\eta_0)\ra \Cal R_0(G)$ of the group $G$ along the geodesic $\a$, $\Cal B_\a(\eta)=F^*_{\eta}$. \endproclaim We notice that deformations of a complex hyperbolic manifold $M$ may depend on many parameters described by the Teichm\"uller space $\Cal T(M)$ of isotopy classes of complex hyperbolic structures on $M$. One can reduce the study of this space $\Cal T(M)$ to studying the variety $\Cal T(G)$ of conjugacy classes of discrete faithful representations $\rho\col G\ra PU(n,1)$ (involving the space $\Cal D(M)$ of the developing maps, see \cite{Go2, FG}). Here $\Cal T(G)=\Cal R_0(G)/PU(n,1)$, and the variety $\Cal R_0(G)\subset \Hom (G, PU(n,1))$ consists of discrete faithful representations $\rho$ of the group $G$ with infinite co-volume, $\Vol (\ch n/G)=\infty$. In particular, our complex bending deformations depend on many independent parameters as it can be shown by applying our construction and \'Elie Cartan \cite {Car} angular invariant in Cauchy-Riemannian geometry: \proclaim{Corollary 7.3} Let $S_p=\rh 2/G$ be a closed totally real geodesic surface of genus $p>1$ in a given complex hyperbolic surface $M=\ch 2/G$, $G\subset PO(2,1)\subset PU(2,1)$. Then there is an embedding $\pi\circ\Cal B\col B^{3p-3}\hra \Cal T(M)$ of a real $(3p-3)$-ball into the Teichm\"uller space of $M$, defined by bending deformations along disjoint closed geodesics in $M$ and by the projection $\pi\col \Cal D(M)\ra \Cal T(M)=\Cal D(M)/PU(2,1)$ in the development space $\Cal D(M)$. \endproclaim \demo{\bf Basic Construction (Proof of Theorem 7.2)} Now we start with a totally real geodesic surface $S=\rh 2/G$ in the complex surface $M=\ch 2/G$, where $G\subset PO(2,1)\subset PU(2,1)$ is a given discrete group, and fix a simple closed geodesic $\a$ on $S$. We may assume that the loop $\a$ is covered by a geodesic $A\subset \rh 2\subset \ch 2$ whose ends at infinity are $\infty$ and the origin of the Heisenberg group $\sch=\bc \times \br$, $\ov \sch =\p \ch 2$. Furthermore, using quasiconformal deformations of the Riemann surface $S$ (in the Teichm\"uller space $\Cal T(S)$, that is, by deforming the inclusion $G\subset PO(2,1)$ in $PO(2,1)$ by bendings along the loop $\a$, see Corollary 3.3 in \cite{A10}), we can assume that the hyperbolic length of $\a$ is sufficiently small and the radius of its tubular neighborhood is big enough: \proclaim{Lemma 7.4} Let $g_\a$ be a hyperbolic element of a non-elementary discrete group $G\subset PO(2,1)\subset PU(2,1)$ with translation length $\ell$ along its axis $A\subset \rh 2$. Then any tubular neighborhood $U_{\da}(A)$ of the axis $A$ of radius $\da>0$ is precisely invariant with respect to its stabilizer $G_0\subset G$ if $ \sinh (\ell/4)%\left(\frac{\ell}{4}\right) \cdot\sinh (\da/2)%\left(\frac{\da}{2}\right) \leq 1/2$. %\frac{1}{2}\,.$$ Furthermore, for sufficiently small $\ell$, $\ell<4\da$, the Dirichlet polyhedron $D_z(G)\subset \ch 2$ of the group $G$ centered at a point $z\in A$ has two sides $a$ and $a'$ intersecting the axis $A$ and such that $g_\a(a)=a'$. \endproclaim Then the group $G$ and its subgroups $G_0, G_1, G_2$ in the free amalgamated (or HNN-extension) decomposition of $G$ have Dirichlet polyhedra $D_z(G_i)\subset \ch 2$,\linebreak $i=0,1,2$, centered at a point $z\in A=(0,\infty)$, whose intersections with the hyperbolic 2-plane $\rh 2$ have the shapes indicated in Figures 2-5. \midinsert $$\hss\psfig{file=bend1.eps,width=2.5truein}\hss \quad \hss\psfig{file=bend2.eps,width=2.5truein}\hss$$ %\vskip1.75truein \botcaption{Figure 2. \,\,\, $G_1\subset G=G_1 *_{G_0}G_2$ \quad Figure 3} %\,\,\, $G_2\subset G=G_1 *_{G_0}G_2$ \endcaption \endinsert %\midinsert %$$\hss\psfig{file=bend2.eps}\hss$$ %\botcaption{Figure 3} %$G_2\subset G=G_1 *_{G_0}G_2$. %\endcaption %\endinsert \midinsert %\vskip1.75truein $$\hss\psfig{file=bend3.eps,width=2.5truein}\hss\quad \hss\psfig{file=bend4.eps,width=2.5truein}\hss$$ \botcaption {Figure 4. \,\,\, $G_1\subset G=G_1 *_{G_0}$ \qquad Figure 5} %\,\,\, $G=G_1 *_{G_0}$ \endcaption \endinsert %\midinsert %$$\hss\psfig{file=bend4.eps}\hss$$ %\botcaption{Figure 5} %$G=G_1 *_{G_0}$. %\endcaption %\endinsert In particular we have that, except two bisectors $\frak S$ and $\frak S'$ that are equivalent under the hyperbolic translation $g_{\a}$ (which generates the stabilizer $G_0\subset G$ of the axis $A$), all other bisectors bounding such a Dirichlet polyhedron lie in sufficiently small ``cone neighborhoods" $C_+$ and $C_-$ of the arcs (infinite rays) $\br_+$ and $\br_-$ of the real circle $\br\times \{0\}\subset \bc\times \br=\sch$. Actually, we may assume that the Heisenberg spheres at infinity of the bisectors $\frak S$ and $\frak S'$ have radii 1 and $r_0>1$, correspondingly. Then, for a sufficiently small $\e$, $0<\e<0$, we can extend the bending homeomorphism $\phi_{\eta,\zeta}$ to an elementary bending homeomorphism $\vp=\vp_{\eta,\zeta}\col\sch\ra\sch$, $\vp(0)=0$, $\vp(\infty)=\infty$, of the whole sphere $S^3=\ov{\sch}$ at infinity. Namely, for the ``dihedral angles" $W_+, W_-\subset\sch$ with the common vertical axis $\{0\}\times \br$ and which are foliated by arcs of real circles connecting points $(0,v)$ and $(0,-v)$ on the vertical axis and intersecting the the $\zeta$-cone neighborhoods of infinite rays $\br_+, \br_- \subset \bc$, correspondingly, the restrictions $\vp|_{W_-}$ and $\vp|_{W_+}$ of the bending homeomorphism $\vp=\vp_{\eta,\zeta}$ are correspondingly the identity and the unitary rotation $U_{\eta}\in PU(2,1)$ by angle $\eta$ about the vertical axis $\{0\}\times\br\subset \sch$, see also \cite{A10, (4.4)}. Then it follows from (7.6) that $\vp_{\eta,\zeta}$ is a $G_0$-equivariant quasiconformal homeomorphism in $\ov{\sch}$. We can naturally extend the foliation of the punctured Heisenberg group $\sch\bs\{0\}$ by Heisenberg spheres $S(0,r)$ to a foliation of the hyperbolic space $\ch 2$ by bisectors $\gtS_r$ having those $S(0,r)$ as the spheres at infinity. It is well known (see \cite{M2}) that each bisector $\gtS_r$ contains a geodesic $\ga_r$ which connects points $(0,-r^2)$ and $(0,r^2)$ of the Heisenberg group $\sch$ at infinity, and furthermore $\gtS_r$ fibers over $\ga_r$ by complex geodesics $Y$ whose circles at infinity are complex circles foliating the sphere $S(0,r)$. Using those foliations of the hyperbolic space $\ch 2$ and bisectors $\gtS_r$, we extend the elementary bending homeomorphism $\vp_{\eta,\zeta}\col\ov{\sch}\ra\ov{\sch}$ at infinity to an elementary bending homeomorphism $\vP_{\eta,\zeta}\col\ov{\ch 2}\ra\ov{\ch 2}$. Namely, the map $\vP_{\eta,\zeta}$ preserves each of bisectors $\gtS_r$, each complex geodesic fiber $Y$ in such bisectors, and fixes the intersection points $y$ of those complex geodesic fibers and the complex geodesic connecting the origin and $\infty$ of the Heisenberg group $\sch$ at infinity. We complete our extension $\vP_{\eta,\zeta}$ by defining its restriction to a given (invariant) complex geodesic fiber $Y$ with the fixed point $y\in Y$. This map is obtained by radiating the circle homeomorphism $\vp_{\eta,\zeta}|_{\p Y}$ to the whole (Poincar\'e) hyperbolic 2-plane $Y$ along geodesic rays $[y,\infty)\subset Y$, so that it preserves circles in $Y$ centered at $y$ and bends (at $y$, by the angle $\eta$) the geodesic in $Y$ connecting the central points of the corresponding arcs of the complex circle $\p Y$, see Fig.6. Due to the construction, the elementary bending (quasiconformal) homeomorphism $\vP_{\eta,\zeta}$ commutes with elements of the cyclic loxodromic group $G_0\subset G$. Another most important property of the homeomorphism $\vP_{\eta,\zeta}$ is the following. Let $D_z(G)$ be the Dirichlet fundamental polyhedron of the group $G$ centered at a given point $z$ on the axis $A$ of the cyclic loxodromic group $G_0\subset G$, and $\gtS^+\subset \ch 2$ be a ``half-space" disjoint from $D_z(G)$ and bounded by a bisector $\gtS\subset \ch 2$ which is different from bisectors $\gtS_r, r>0$, and contains a side $\gts$ of the polyhedron $D_z(G)$. Then there is an open neighborhood $U(\ov{\gtS^+})\subset \ov{\ch 2}$ such that the restriction of the elementary bending homeomorphism $\vP_{\eta,\zeta}$ to it either is the identity or coincides with the unitary rotation $U_{\eta}\subset PU(2,1)$ by the angle $\eta$ about the ``vertical" complex geodesic (containing the vertical axis $\{0\}\times\br\subset \sch$ at infinity). The above properties of quasiconformal homeomorphism $\vP=\vP_{\eta,\zeta}$ show that the image $D_{\eta}=\vP_{\eta,\zeta}(D_z(G))$ is a polyhedron in $\ch 2$ bounded by bisectors. Furthermore, there is a natural identification of its sides induced by $\vP_{\eta,\zeta}$. Namely, the pairs of sides preserved by $\vP$ are identified by the original generators of the group $G_1\subset G$. For other sides $\gts_{\eta}$ of $D_{\eta}$, which are images of corresponding sides $\gts\subset D_z(G)$ under the unitary rotation $U_{\eta}$, we define side pairings by using the group $G$ decomposition (see Fig. 2-5). Actually, if $G=G_1 *_{G_0}G_2$, we change the original side pairings $g\in G_2$ of $D_z(G)$-sides to the hyperbolic isometries $U_{\eta}gU_{\eta}^{-1}\in PU(2,1)$. In the case of HNN-extension, $G=G_1 *_{G_0}=\langle G_1, g_2\rangle$, we change the original side pairing $g_2\in G$ of $D_z(G)$-sides to the hyperbolic isometry $U_{\eta}g_2\in PU(2,1)$. In other words, we define deformed groups $G_{\eta}\subset PU(2,1)$ correspondingly as $$ G_{\eta}=G_1 *_{G_0}U_{\eta}G_2U^{-1}_{\eta}\quad \text{or}\quad G_{\eta}=\langle G_1, U_{\eta}g_2\rangle=G_1 *_{G_0}\,.\tag7.7 $$ This shows that the family of representations $G\ra G_{\eta}\subset PU(2,1)$ does not depend on angles $\zeta$ and holomorphically depends on the angle parameter $\eta$. Let us also observe that, for small enough angles $\eta$, the behavior of neighboring polyhedra $g'(D_{\eta})$, $g'\in G_{\eta}$ is the same as of those $g(D_z(G))$, $g\in G$, around the Dirichlet fundamental polyhedron $D_z(G)$. This is because the new polyhedron $D_{\eta}\subset \ch 2$ has isometrically the same (tesselations of) neighborhoods of its side-intersections as $D_z(G)$ had. This implies that the polyhedra $g'(D_{\eta})$, $g'\in G_{\eta}$, form a tesselation of $\ch 2$ (with non-overlapping interiors). Hence the deformed group $G_{\eta}\subset PU(2,1)$ is a discrete group, and $D_{\eta}$ is its fundamental polyhedron bounded by bisectors. Using $G$-compatibility of the restriction of the elementary bending homeomorphism $\vP=\vP_{\eta,\zeta}$ to the closure $\ov{D_z(G)}\subset \ov{\ch 2}$, we equivariantly extend it from the polyhedron $\ov{D_z(G)}$ to the whole space $\ch 2\cup\Om(G)$ accordingly to the $G$-action. In fact, in terms of the natural isomorphism $\chi\col G\ra G_{\eta}$ which is identical on the subgroup $G_1\subset G$, we can write the obtained $G$-equivariant homeomorphism $F=F_{\eta}\,\col\, \ov{\ch 2}\bs\La(G)\ra\ov{\ch 2}\bs\La(G_{\eta})$ in the following form: $$ \aligned F_{\eta}(x)&=\vP_{\eta}(x)\quad\text{for }\quad x\in \ov{D_z(G)},\\ F_{\eta}\circ g(x)&=g_{\eta}\circ F_{\eta}(x) \quad\text{for }\quad x\in \ov{\ch 2}\bs\La(G),\,\, g\in G,\,\, g_{\eta}=\chi(g)\in G_{\eta}\,. \endaligned\tag7.8 $$ Due to quasiconformality of $\vP_{\eta}$, the extended $G$-equivariant homeomorphism $F_{\eta}$ is quasiconformal. Furthermore, its extension by continuity to the limit (real) circle $\La(G)$ coincides with the canonical equivariant homeomorphism $f_{\chi}\col \La(G)\ra\La(G_{\eta})$ given by the isomorphism Theorem 6.2. Hence we have a $G$-equivariant quasiconformal self-homeomorphism of the whole space $\ov{\ch 2}$, which we denote as before by $F_{\eta}$. The family of $G$-equivariant quasiconformal homeomorphisms $F_{\eta}$ induces representations $F^*_{\eta}\,\col\, G\to G_{\eta}=F_{\eta}G_2F^{-1}_{\eta}$, $\eta\in (-\eta_0,\,\eta_0)$. In other words, we have a curve $\Cal B\col (-\eta_0,\,\eta_0)\ra \Cal R_0(G)$ in the variety $\Cal R_0(G)$ of faithful discrete representations of $G$ into $PU(2,1)$, which covers a nontrivial curve in the Teichm\"uller space $\Cal T(G)$ represented by conjugacy classes $[\Cal B(\eta)]=[F^*_{\eta}]$. We call the constructed deformation $\Cal B$ the bending deformation of a given lattice $G\subset PO(2,1)\subset PU(2,1)$ along a bending geodesic $A\subset \ch 2$ with loxodromic stabilizer $G_0\subset G$. In terms of manifolds, $\Cal B$ is the bending deformation of a given complex surface $M=\ch 2/G$ homotopy equivalent to its totally real geodesic surface $S_g\subset M$, along a given simple geodesic $\a$. \line{\hfil \hfil \qed} \enddemo \remark{Remark 7.9} It follows from the above construction of the bending homeomorphism $F_{\eta, \zeta}$, that the deformed complex hyperbolic surface $M_{\eta}=\ch 2/G_{\eta}$ fibers over the pleated hyperbolic surface $S_{\eta}=F_{\eta}(\rh 2)/G_{\eta}$ (with the closed geodesic $\a$ as the singular locus). The fibers of this fibration are ``singular real planes" obtained from totally real geodesic 2-planes by bending them by angle $\eta$ along complete real geodesics. These (singular) real geodesics are the intersections of the complex geodesic connecting the axis $A$ of the cyclic group $G_0\subset G$ and the totally real geodesic planes that represent fibers of the original fibration in $M=\ch 2/G$. \endremark \demo{Proof of Corollary 7.3} Since, due to (7.7), bendings along disjoint closed geodesics are independent, we need to show that our bending deformation is not trivial, and $[\Cal B(\eta)]\neq [\Cal B(\eta')]$ for any $\eta\neq\eta'$. The non-triviality of our deformation follows directly from (7.7), cf. \cite{A9}. Namely, the restrictions $\rho_{\eta}|_{G_1}$ of bending representations to a non-elementary subgroup $G_1\subset G$ (in general, to a ``real" subgroup $G_r\subset G$ corresponding to a totally real geodesic piece in the homotopy equivalent surface $S\backsimeq M$) are identical. So if the deformation $\Cal B$ were trivial then it would be conjugation of the group $G$ by projective transformations that commute with the non-trivial real subgroup $G_r\subset G$ and pointwise fix the totally real geodesic plane $\rh 2$. This contradicts to the fact that the limit set of any deformed group $G_{\eta}$, $\eta\neq 0$, does not belong to the real circle containing the limit Cantor set $\La(G_r)$. The injectivity of the map $\Cal B$ can be obtained by using \'Elie Cartan \cite{Car} angular invariant $\ba(x)$, $-\pi/2\leq\ba(x)\leq\pi/2$, for a triple $x=(x^0, x^1, x^2)$ of points in $\p\ch 2$. It is known (see \cite{Go3}) that, for two triples $x$ and $y$, $\ba(x)=\ba(y)$ if and only if there exists $g\in PU(2,1)$ such that $y=g(x)$; furthermore, such a $g$ is unique provided that $\ba(x)$ is neither zero nor $\,\pm \pi/2$. Here $\ba(x)=0$ if and only if $x^0, x^1$ and $x^2$ lie on an $\br$-circle, and $\ba(x)=\pm \pi/2$ if and only if $x^0, x^1$ and $x^2$ lie on a chain ($\bc$-circle). Namely, let $g_2\in G\bs G_1$ be a generator of the group $G$ in (4.5) whose fixed point $x^2\in \La(G)$ lies in $\br_+\times \{0\} \subset \sch$, and $x^2_{\eta}\in \La(G_{\eta})$ the corresponding fixed point of the element $\chi_{\eta}(g_2)\in G_{\eta}$ under the free-product isomorphism $\chi_{\eta}\col G\ra G_{\eta}$. Due to our construction, one can see that the orbit $\ga(x^2_{\eta})$, $\ga\in G_0$, under the loxodromic (dilation) subgroup $G_0\subset G\cap G_{\eta}$ approximates the origin along a ray $(0,\infty)$ which has a non-zero angle $\eta$ with the ray $\br_-\times \{0\}\subset \sch$. The latter ray also contains an orbit $\ga(x^1)$, $\ga\in G_0$, of a limit point $x^1$ of $G_1$ which approximates the origin from the other side. Taking triples $x=(x^1, 0, x^2)$ and $x_{\eta}=(x^1, 0, x^2_{\eta})$ of points which lie correspondingly in the limit sets $\La(G)$ and $\La(G_{\eta})$, we have that $\ba(x)=0$ and $\ba(x_{\eta})\neq 0, \,\pm\pi/2$. Due to Theorem 6.2, both limit sets are topological circles which however cannot be equivalent under a hyperbolic isometry because of different Cartan invariants (and hence, again, our deformation is not trivial). Similarly, for two different values $\eta$ and $\eta'$, we have triples $x_{\eta}$ and $x_{\eta'}$ with different (non-trivial) Cartan angular invariants $\ba(x_{\eta})\neq \ba(x_{\eta'})$. Hence $\La(G_{\eta})$ and $\La(G_{\eta'})$ are not $PU(2,1)$-equivalent. \line{\hfil \hfil \qed} \enddemo One can apply the above proof to a general situation of bending deformations of a complex hyperbolic surface $M=\ch 2/G$ whose holonomy group $G\subset PU(2,1)$ has a non-elementary subgroup $G_r$ preserving a totally real geodesic plane $\rh 2$. In other words, such a complex surfaces $M$ has an embedded totally real geodesic surface with geodesic boundary. In particular all complex surfaces constructed in \cite{GKL} with a given Toledo invariant lie in this class. So we immediately have: \proclaim{Corollary 7.10} Let $M=\ch 2/G$ be a complex hyperbolic surface with embedded totally real geodesic surface $S_r\subset M$ with geodesic boundary, and $\Cal B\col (-\eta, \eta)\ra \Cal D(M)$ be the bending deformation of $M$ along a simple closed geodesic $\a\subset S_r$. Then the map $\pi\circ\Cal B\col (-\eta, \eta)\ra \Cal T(M)=\Cal D(M)/PU(2,1)$ is a smooth embedding provided that the limit set $\La(G)$ of the holonomy group $G$ does not belong to the $G$-orbit of the real circle $S^1_{\br}$ and the chain $S^1_{\bc}$, where the latter is the infinity of the complex geodesic containing a lift $\tilde\a\subset\ch 2$ of the closed geodesic $\a$, and the former one contains the limit set of the holonomy group $G_r\subset G$ of the geodesic surface $S_r$. \line{\hfil\hfil\qed} \endproclaim As an application of the constructed bending deformations, we answer a well known question about cusp groups on the boundary of the Teichm\"uller space $\sct (M)$ of a Stein complex hyperbolic surface $M$ fibering over a compact Riemann surface of genus $p>1$. It is a direct corollary of the following result, see \cite{AG}: \proclaim {Theorem 7.11} Let $G\subset PO(2,1)\subset PU(2,1)$ be a non-elementary discrete group $S_p$ of genus $p\geq 2$. Then, for any simple closed geodesic $\a$ in the Riemann surface $S=H^2_{\br}/G$, there is a continuous deformation $\rho_t=f^*_t$ induced by $G$-equivariant quasiconformal homeomorphisms $f_t: \ov{\ch 2} \to \ov{\ch 2}$ whose limit representation $\rho_{\infty}$ corresponds to a boundary cusp point of the Teichm\"uller space $\Cal T(G)$, that is, the boundary group $\rho_{\infty}(G)$ has an accidental parabolic element $\rho_{\infty}(g_{\a})$ where $g_{\a}\in G$ represents the geodesic $\a\subset S$. \endproclaim We note that, due to our construction of such continuous quasiconformal deformations in \cite {AG}, they are independent if the corresponding geodesics $\a_i\subset S_p$ are disjoint. It implies the existence of a boundary group in $\p \Cal T(G)$ with ``maximal" number of non-conjugate accidental parabolic subgroups: \proclaim {Corollary 7.12} Let $G\subset PO(2,1)\subset PU(2,1)$ be a uniform lattice isomorphic to the fundamental group of a closed surface $S_p$ of genus $p\geq 2$. Then there is a continuous deformation $R\col \br^{3p-3}\to \Cal T(G)$ whose boundary group $G_{\infty}=R(\infty)(G)$ has $(3p-3)$ non-conjugate accidental parabolic subgroups. \endproclaim Finally, we mention another aspect of the intrigue Problem 4.12 on geometrical finiteness of complex hyperbolic surfaces (see \cite {AX1, AX2}) for which it may perhaps be possible to apply our complex bending deformations: \proclaim {Problem} Construct a geometrically infinite (finitely generated) discrete group $G\subset PU(2,1)$ whose limit set is the whole sphere at infinity, $\La(G)=\p \ch 2=\ov {\sch}$, and which is the limit of convex cocompact groups $G_i\subset PU(2,1)$ from the Teichm\"uller space $\Cal T(\Ga)$ of a convex cocompact group $\Ga\subset PU(2,1)$. Is that possible for a Schottky group $\Ga$? \endproclaim %\vfil %\eject \def\ref#1{[#1]} \eightpoint \parindent=36pt \head REFERENCES \endhead \bigskip \frenchspacing \item{\ref{Ah}} Lars V. 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