Geometry.and.topology.of.complex.hyperbolic.and.CR-manifolds

GEOMETRY AND TOPOLOGY OF COMPLEX

HYPERBOLIC AND CR�MANIFOLDS

Boris Apanasov

ABSTRACT� We study geometry� topology and deformation spaces of noncompactcomplex hyperbolic manifolds �geometrically �nite� with variable negative curvature��whose properties make them surprisingly di�erent from real hyperbolic manifolds

with constant negative curvature� This study uses an interaction between K�ahlergeometry of the complex hyperbolic space and the contact structure at its in�nity�the one�point compacti�cation of the Heisenberg group�� in particular an establishedstructural theorem for discrete group actions on nilpotent Lie groups�

�� Introduction

This paper presents recent progress in studying topology and geometry of com�plex hyperbolic manifoldsM with variable negative curvature and spherical Cauchy�Riemannian manifolds with Carnot�Caratheodory structure at in�nity M��Among negatively curved manifolds� the class of complex hyperbolic manifolds

occupies a distinguished niche due to several reasons� First� such manifolds fur�nish the simplest examples of negatively curved K�ahler manifolds� and due to theircomplex analytic nature� a broad spectrum of techniques can contribute to thestudy� Simultaneously� the in�nity of such manifolds� that is the spherical Cauchy�Riemannian manifolds furnish the simplest examples of manifolds with contactstructures� Second� such manifolds provide simplest examples of negatively curvedmanifolds not having constant sectional curvature� and already obtained resultsshow surprising di�erences between geometry and topology of such manifolds andcorresponding properties of �real hyperbolic� manifolds with constant negative cur�vature� see �BS� BuM� EMM� Go� GM� Min� Yu� Third� such manifolds occupy aremarkable place among rank�one symmetric spaces in the sense of their deforma�tions� they enjoy the �exibility of low dimensional real hyperbolic manifolds �see�Th� A� A and x�� as well as the rigidity of quaternionic�octionic hyperbolicmanifolds and higher�rank locally symmetric spaces �MG� Co � P� Finally� since

�� Mathematics Subject Classi�cation� �� Key words and phrases� Complex hyperbolic geometry� Cauchy�Riemannian manifolds� dis�

crete groups� geometrical �niteness� nilpotent and Heisenberg groups� Bieberbach theorems� �berbundles� homology cobordisms� quasiconformal maps� structure deformations� Teichm�uller spaces�Research in MSRI was supported in part by NSF grant DMS��

Typeset by AMS�TEX

�

BORIS APANASOV

its inception� the theory of smooth ��manifolds has relied upon complex surfacetheory to provide its basic examples� Nowadays it pays back� and one can studycomplex analytic �manifolds by using Seiberg�Witten invariants� decomposition of��manifolds along homology ��spheres� Floer homology and new �homology� cobor�dism invariants� see �W� LB� BE� FS� S� A� and x��Complex hyperbolic geometry is the geometry of the unit ball B n

Cin C n with the

K�ahler structure given by the Bergman metric �compare �CG� Go�� whose auto�morphisms are biholomorphic automorphisms of the ball� i�e�� elements of PU�n� ��We notice that complex hyperbolic manifolds with non�elementary fundamentalgroups are complex hyperbolic in the sense of S�Kobayashi �Kob�� Here we studytopology and geometry of complex hyperbolic manifolds by using spherical Cauchy�Riemannian geometry at their in�nity� This CR�geometry is modeled on the onepoint compacti�cation of the �nilpotent� Heisenberg group� which appears as thesphere at in�nity of the complex hyperbolic space H n

C� In particular� our study

exploits a structural Theorem �� about actions of discrete groups on nilpotent Liegroups �in particular on the Heisenberg group Hn�� which generalizes a Bieberbachtheorem for Euclidean spaces �Wo and strengthens a result by L�Auslander �Au�Our main assumption on a complex hyperbolic n�manifold M is the geometrical

�niteness condition on its fundamental group ��M� � G � PU�n� �� which inparticular implies that G is �nitely generated �Bow and even �nitely presented� seeCorollary �� The original de�nition of a geometrically �nite manifold M �due toL�Ahlfors � Ah� came from an assumption that M may be decomposed into a cellby cutting along a �nite number of its totally geodesic hypersurfaces� The notionof geometrical �niteness has been essentially used in the case of real hyperbolicmanifolds �of constant sectional curvature�� where geometric analysis and ideas ofThurston have provided powerful tools for understanding of their structure� see�BM� MA� Th� A� A�� Some of those ideas also work in spaces with pinched neg�ative curvature� see �Bow� However� geometric methods based on consideration of�nite sided fundamental polyhedra cannot be used in spaces of variable curvature�see x�� and we base our geometric description of geometrically �nite complex hyper�bolic manifolds on a geometric analysis of their �thin� ends� This analysis is basedon establishing a �ber bundle structure on Heisenberg �in general� non�compact�manifolds which remind Gromov�s almost �at �compact� manifolds� see �Gr� BK�As an application of our results on geometrical �niteness� we are able to �nd

�nite coverings of an arbitrary geometrically �nite complex hyperbolic manifoldsuch that their parabolic ends have the simplest possible structure� i�e�� ends witheither Abelian or �step nilpotent holonomy �Theorem �� In another such an ap�plication� we study an interplay between topology and K�ahler geometry of complexhyperbolic n�manifolds� and topology and Cauchy�Riemannian geometry of theirboundary � n� ��manifolds at in�nity� see our homology cobordism Theorem ��In that respect� the problem of geometrical �niteness is very di�erent in complex di�mension two� where it is quite possible that complex surfaces with �nitely generatedfundamental groups and �big� ends at in�nity are in fact geometrically �nite� Wealso note that such non�compact geometrically �nite complex hyperbolic surfaces

COMPLEX HYPERBOLIC AND CR�MANIFOLDS �

have in�nitely many smooth structures� see �BE�The homology cobordism Theorem �� is also an attempt to control the bound�

ary components at in�nity of complex hyperbolic manifolds� Here the situationis absolutely di�erent from the real hyperbolic one� In fact� due to Kohn�Rossianalytic extension theorem in the compact case �EMM and to D�Burns theorem inthe case when only one boundary component at in�nity is compact �see also �NR�Th�� NR �� the whole boundary at in�nity of a complex hyperbolic manifoldM of in�nite volume is connected �and the manifold itself is geometrically �nite ifdimC M � �� if one of the above compactness conditions holds� However� if bound�ary components of M are non�compact� the boundary ��M may have arbitrarilymany components due to our construction in Theorems �� and ��The results on geometrical �niteness are naturally linked with the Sullivan�s

stability of discrete representations of ��M� into PU�n� �� deformations of com�plex hyperbolic manifolds and Cauchy�Riemannian manifolds at their in�nity� andequivariant �quasiconformal or quasisymmetric� homeomorphisms inducing suchdeformations and isomorphisms of discrete subgroups of PU�n� �� Results in thesedirections 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 �exibility� Namely� �nite volumecomplex hyperbolic manifolds are rigid due to Mostow�s rigidity �Mo �for all locallysymmetric spaces of rank one�� Nevertheless their constant curvature analogue�real hyperbolic manifolds are �exible in low dimensions and in the sense of quasi�Fuchsian deformations �see our discussion in x�� Contrasting to such a �exibility�complex hyperbolic manifolds share the super�rigidity of quaternionic�octionic hy�perbolic manifolds �see Pansu�s �P and Corlette�s �Co� rigidity theorems� analo�gous to Margulis�s �MG super�rigidity in higher rank�� Namely� due to Goldman�s�Go local rigidity theorem in dimension n � and its extension �GM for n � ��every nearby discrete representation � � G � PU�n� � of a cocompact latticeG � PU�n� � � stabilizes a complex totally geodesic subspace H n��

Cin H n

C� and

for n � �� this rigidity is even global due to a celebrated Yue�s theorem �Yu�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 rigidin general� Such a �exibility has two aspects� Firstly� we point out that the rigid�ity condition that the group G � PU�n� � preserves a complex totally geodesichyperspace in H n

Cis essential for local rigidity of deformations only for co�compact

lattices G � PU�n� � �� This is due to the following our result �ACG�

Theorem �� Let G � PU�� be a co��nite free lattice whose action in H �C

is generated by four real involutions �with �xed mutually tangent real circles atin�nity�� Then there is a continuous family ff�g� �� of G�equivariant

homeomorphisms in H �Cwhich induce non�trivial quasi�Fuchsian �discrete faithful�

representations f�� G � PU� � �� Moreover� for each � �� any G�equivariant

homeomorphism of H �Cthat induces the representation f�� cannot be quasiconformal�

This also shows the impossibility to extend the Sullivan�s quasiconformal stability

� BORIS APANASOV

theorem �Su to that situation� as well as provides the �rst continuous �topological�deformation of a co��nite Fuchsian group G � PU�� into quasi�Fuchsian groupsG� � f�Gf

�� PU� � � with the �arbitrarily close to one� Hausdor� dimension

dimH ��G�� of the limit set ��G�� compare �Co�Secondly� we point out that the noncompactness condition in our non�rigidity

theorem is not essential� either� Namely� complex hyperbolic Stein manifolds ho�motopy equivalent to their closed totally real geodesic surfaces are not rigid� too�Namely� in complex dimension n � � we provide a canonical construction of con�tinuous quasi�Fuchsian deformations of complex surfaces �bered over closed Rie�mannian surfaces� which we call �complex bendings� along simple close geodesics�This is the �rst such deformations �moreover� quasiconformally induced ones� ofcomplex analytic �brations over a compact base�

Theorem �� Let G � PO� � � � PU� � � be a given �non�elementary� discretegroup� Then� for any simple closed geodesic � in the Riemann ��surface S � H�

R�G

and a su�ciently small � � �� there is a holomorphic family of G�equivariant

quasiconformal homeomorphisms F� � H �C� H �

C� �� which de�nes

the bending �quasi�Fuchsian� deformation B� � �� R��G� of the group Galong the geodesic �� B�� F �

� �

The constructed deformations depend on many parameters described by theTeichm�uller space T �M� of isotopy classes of complex hyperbolic structures on M �or equivalently by the Teichm�uller space T �G� � R��G��PU�n� � of conjugacyclasses of discrete faithful representations � � R��G� � Hom�G�PU�n� �� of G ��M��

Corollary �� Let Sp � H�R�G be a closed totally real geodesic surface of genus

p � in a given complex hyperbolic surface M � H �C�G� G � PO� � � � PU� � ��

Then there is an embedding � � B � B�p�� T �M� of a real ��p � ��ball intothe Teichm�uller space of M � de�ned by bending deformations along disjoint closedgeodesics in M and by the projection � � D�M�� T �M� � D�M��PU� � � in thedevelopment space D�M��

As an application of the constructed deformations� we answer a well knownquestion about cusp groups on the boundary of the Teichm�uller space T �M� of a�Stein� complex hyperbolic surface M �bering over a compact Riemann surface ofgenus p � �AG�

Corollary �� Let G � PO� � � � PU� � � be a uniform lattice isomorphicto the fundamental group of a closed surface Sp of genus p � � Then there is acontinuous deformation R � R�p�� T �G� �induced by G�equivariant quasiconfor�mal homeomorphisms of H �

C� whose boundary group G� � R��G� has ��p� ��

non�conjugate accidental parabolic subgroups�

Naturally� all constructed topological deformations are in particular geometricrealizations of the corresponding �type preserving� discrete group isomorphisms�

COMPLEX HYPERBOLIC AND CR�MANIFOLDS

see Problem �� However� as Example �� shows� not all such type preserving iso�morphisms are so good� Nevertheless� as the �rst step in solving the geometrizationProblem �� we prove the following geometric realization theorem �A��

Theorem �� Let � � G � H be a type preserving isomorphism of two non�ele�mentary geometrically �nite groups G�H � PU�n� �� Then there exists a uniqueequivariant homeomorphism f� � ��G�� H� of their limit sets that induces theisomorphism �� Moreover� if ��G� � �H n

C� the homeomorphism f� is the restriction

of a hyperbolic isometry h � PU�n� ��

We note that� in contrast to Tukia �Tu isomorphism theorem in the real hyper�bolic geometry� one might suspect that in general the homeomorphism f� has nogood metric properties� compare Theorem �� This is still one of open problemsin complex hyperbolic geometry �see x� for discussions��

�� Complex hyperbolic and Heisenberg manifolds

We recall some facts concerning the link between nilpotent geometry of theHeisenberg group� the Cauchy�Riemannian geometry �and contact structure� of itsone�point compacti�cation� and the K�ahler geometry of the complex hyperbolicspace �compare �GP� Go�� KR��One can realize the complex hyperbolic geometry in the complex projective space�

HnC � f�z � C P

n � hz� zi � � � z � Cn��g �

as the set of negative lines in the Hermitian vector space Cn�� with Hermitian

structure given by the inde�nite �n� ��form hz� wi � z�w�� znwn�zn��wn��Its boundary �H n

C� f�z � C Pn�� hz� zi � �g consists of all null lines in C Pn and

is homeomorphic to the � n��sphere S�n��The full group Isom H n

Cof isometries of H n

Cis generated by the group of holo�

morphic automorphisms �� the projective unitary group PU�n� � de�ned by thegroup U�n� � of unitary automorphisms of C n�� together with the antiholomor�phic automorphism of H n

Cde�ned by the C �antilinear unitary automorphism of C n��

given by complex conjugation z � �z� The group PU�n� � can be embedded in alinear group due to A�Borel �Bor �cf� �AX� L� �� hence any �nitely generatedgroup G � PU�n� � is residually �nite and has a �nite index torsion free subgroup�Elements g � PU�n� � are of the following three types� If g �xes a point in H n

C� it

is called elliptic� If g has exactly one �xed point� and it lies in �H nC� g is called par�

abolic� If g has exactly two �xed points� and they lie in �H nC� g is called loxodromic�

These three types exhaust all the possibilities�There are two common models of complex hyperbolic space H n

Cas domains in

C n � the unit ball B nCand the Siegel domain Sn� They arise from two a�ne patches

in projective space related to H nCand its boundary� Namely� embedding C n onto

the a�ne patch of C Pn�� de�ned by zn�� in homogeneous coordinates� asA � C n � C Pn � z � ��z� �� we may identify the unit ball B n

C�� C n with H n

C�

A�B nC�� Here the metric in C n is de�ned by the standard Hermitian form hh � ii�

� BORIS APANASOV

and the induced metric on BnCis the Bergman metric �with constant holomorphic

curvature �� whose sectional curvature is between � and ��The Siegel domain model of H n

Carises from the a�ne patch complimentary to

a projective hyperplane H� which is tangent to �H nCat a point � � �H n

C� For

example� taking that point� as �� with �� C n�� and H� � f�z � C Pn �zn � zn�� g� one has the map S � C n � C PnnH� such that

�z�

zn

��

�� z�

�� zn

�� zn

�� where z� �

��

z��

zn��

�A � C

n��

In the obtained a�ne coordinates� the complex hyperbolic space is identi�edwith the Siegel domain

Sn � S��H nC � � fz � C

n � zn � zn � hhz�� z�iig �

where the Hermitian form is hS�z��S�w�i � hhz�� w�ii � zn � wn� The automor�phism group of this a�ne model of H n

Cis the group of a�ne transformations of C n

preserving Sn� Its unipotent radical is the Heisenberg group Hn consisting of allHeisenberg translations

T��v � �w�� wn� �

�w� � � wn � hh � w

�ii�

�hh � xiii � iv�

��

where w�� C n�� and v � R�In particularHn acts simply transitively on �H

nCnf�g� and one obtains the upper

half space model for complex hyperbolic space H nCby identifying C n�� R ��

and H nCnf�g as

� � v� u� ��

��

�� hh � ii � u� iv�� hh � ii� u� iv�

��

where � � v� u� � Cn�� R �� are the horospherical coordinates of the corre�

sponding point in H nCnf�g �with respect to the point � � �H n

C� see �GP��

We notice that� under this identi�cation� the horospheres in H nCcentered at �

are the horizontal slices Ht � f� � v� u� � C n�� R R� � u � tg� and thegeodesics running to � are the vertical lines c��v�t� � � � v� e�t� passing throughpoints � � v� � C n�� R� Thus we see that� via the geodesic perspective from ��various horospheres correspond as Ht � Hu with � � v� t� � � � v� u��The �boundary plane� C n�� R f�g � �H n

Cnf�g and the horospheres Hu �

Cn��Rfug� � � u �� centered at� are identi�ed with the Heisenberg group

Hn � C n�� R� It is a �step nilpotent group with center f�g R � C n�� R�with the isometric action on itself and on H n

Cby left translations�

T��v�� v� u� �� v� � v � Imhh �� ii � u� �


and the inverse of � � v� is � � v�� v�� The unitary group U�n� � acts onHn and H n

Cby rotations� A� � v� u� � �A � v � u� for A � U�n� �� The semidirect

product H�n� � Hn o U�n� � is naturally embedded in U�n� � as follows�

A ��

��A � ��

�A � U�n� � for A � U�n� � �

� � v� ��

�� In�� t � �

��j j� � iv� ��

� �j j� � iv�

� t �� j j

� � iv� � �� j j

� � iv�

�A � U�n� �

where � � v� � Hn � C n�� R and � t is the conjugate transpose of �The action of H�n� on H n

Cnf�g also preserves the Cygan metric �c there� which

plays the same role as the Euclidean metric does on the upper half�space model ofthe real hyperbolic space H n and is induced by the following norm�

jj� � v� u�jjc � j jj jj� � u� ivj�� v� u� � Cn�� R ��

The relevant geometry on each horosphere Hu � H nC� Hu

�� Hn � C n�� R�is the spherical CR�geometry induced by the complex hyperbolic structure� Thegeodesic perspective from � de�nes CR�maps between horospheres� which extendto CR�maps between the one�point compacti�cations Hu � � S�n�� In thelimit� the induced metrics on horospheres fail to converge but the CR�structureremains �xed� In this way� the complex hyperbolic geometry induces CR�geometryon the sphere at in�nity �H n

C S�n�� naturally identi�ed with the one�point

compacti�cation of the Heisenberg group Hn�

�� Discrete actions on nilpotent groups and Heisenberg manifolds

In order to study the structure of Heisenberg manifolds �i�e�� the manifolds lo�cally modeled on the Heisenberg group Hn� and cusp ends of complex hyperbolicmanifolds� we need a Bieberbach type structural theorem for isometric discretegroup actions on Hn� originally proved in �AX� It claims that each discrete isom�etry group of the Heisenberg group Hn preserves some left coset of a connected Liesubgroup� on which the group action is cocompact�Here we consider more general situation� Let N be a connected� simply con�

nected nilpotent Lie group� C a compact group of automorphisms of N � and �a discrete subgroup of the semidirect product N o C� Such discrete groups arethe holonomy groups of parabolic ends of locally symmetric rank one �negativelycurved� manifolds and can be described as follows�

Theorem �� There exist a connected Lie subgroup V of N and a �nite indexnormal subgroup �� of � with the following properties

�� There exists b � N such that b�b�� preserves V �� V�b�b�� is compact�� b��b�� acts on V by left translations and this action is free�

� BORIS APANASOV

Remark �� It immediately follows that any discrete subgroup � � N o C isvirtually nilpotent because it has a �nite index subgroup �� isomorphic to alattice in V � N �� Here� compactness of C is an essential condition because of Margulis �MG

construction of nonabelian free discrete subgroups � of R� oGL�� R�� This theorem generalizes a Bieberbach theorem for Euclidean spaces� see

�Wo� and strengthens a result by L�Auslander �Au who claimed those propertiesnot for whole group � but only for its �nite index subgroup� Initially in �AX� weproved this theorem for the Heisenberg group Hn where we used Margulis Lemma�MG� BGS and geometry of Hn in order to extend the classical arguments in �Wo�In the case of general nilpotent groups� our proof uses di�erent ideas and goes asfollows �see�AX for details��

Sketch of Proof� Let p � � � C be the composition of the inclusion � � N o Cand the projection N oC � C� G the identity component of �N � and �� G� ��Due to compactness of C� G has �nite index in �N � so �� has �nite index in ��Let W � N be the analytic subgroup pointwise �xed by p�� Due to �Au� for all� � �w� c� � �� w lies in W � Thus � preserves W and� by replacing N with W � wemay assume that � p�� is �nite�Consider �� ker�p� which is a discrete subgroup of N and has �nite index in

�� Let V be the connected Lie subgroup of N in which �� is a lattice� Then theconjugation action of � on �� induces a ��action on V � We form the semi�directproduct V o � and let K � f�a�� a� �� V o � � �a� � � ��g� Obviously�K is a normal subgroup of V o �� De�ning the maps i � V � V o ��K byi�v� � �v� �� K and � � V o ��K � by ��v� �a�A�� A� we get a short exactsequence

�� Vi

�� V o ��K�

��

Since any extension of a �nite group by a simply connected nilpotent Lie groupsplits� there is a homomorphism s � � V o ��K such that � � s � id�� For eachA � � we �x an element �f�A�� g�A�� A�� V o � representing s�A�� Since s is ahomomorphism� we have

g�AB��f�AB�� Ag�B��f�B��

g�A��f�A�� for A�B � � ��

De�ne h � � N by h�A� � g�A��f�A�� Then � �� shows that h isa cocycle� Since is �nite and N is a simply connected nilpotent Lie group�H�� N� � � due to �LR� Thus there exists b � N such that h�A� � A�b��b forall A � �On the other hand� �� a�A��K� � ��f�A�� g�A�� A��K� � A for any � �

�a�A� � �� It follows that there is v� � V such that a��v� � h�A�� This and ��imply that a��v� � A�b��b� and hence baA�b�� bv�b

��Now consider the group b�b�� which acts on bV b�� For any � � �a�A� � ��

the action of the element b�b�� baA�b�� A� on bV b�� is as follows�

��baA�b�� A�� v�� baA�b��A�v��baA�b��


In particular� baA�b��A�bV b��baA�b�� bV b�� Therefore� A�bV b�� bV b�� because of baA�b�� bv�b

�� bV b�� and hence b�b�� preserves bV b��

Now we can apply our description of discrete group actions on a nilpotent group�Theorem �� to study the structure of Heisenberg manifolds� Such manifolds arelocally modeled on the �Hn�H�n��geometry and each of them can be representedas the quotient Hn�G under a discrete� free isometric action of its fundamentalgroup G on Hn� i�e�� the isometric action of a torsion free discrete subgroup ofH�n� � Hn o U�n � �� Actually� we establish �ber bundle structures on allnoncompact Heisenberg manifolds�

Theorem �� Let � � Hn o U�n� � be a torsion�free discrete group acting onthe Heisenberg group Hn � C n��R with non�compact quotient� Then the quotientHn�� has zero Euler characteristic and is a vector bundle over a compact manifold�Furthermore� this compact manifold is �nitely covered by a nil�manifold which iseither a torus or the total space of a circle bundle over a torus�

The proof of this claim �see �AX� is based on two facts due to Theorem ��First� that the discrete holonomy group � �� M� of any noncompact HeisenbergmanifoldM � Hn�� H�n�� has a proper ��invariant subspace H� � Hn� Andsecond� the compact manifold H�� is �nitely covered by H��

� where �� acts onH� by translations� The structure of the covering manifold HG�G

� is given in thefollowing lemma�

Lemma �� Let V be a connected Lie subgroup of the Heisenberg group Hn andG � V a discrete co�compact subgroup of V � Then the manifold V�G is

�� a torus if V is Abelian�� the total space of a torus bundle over a torus if V is not Abelian�

Though noncompact Heisenberg manifoldsM are vector bundlesHn��H��simple examples show �AX that such vector bundles may be non�trivial in general�However� up to �nite coverings� they are trivial �AX�

Theorem �� Let � � HnoU�n�� be a discrete group and H� � Hn a connected��invariant Lie subgroup on which � acts co�compactly� Then there exists a �niteindex subgroup �� such that the vector bundle Hn�� H�� is trivial�In particular� any Heisenberg orbifold Hn�� is �nitely covered by the product of acompact nil�manifold H�� and an Euclidean space�

We remark that in the case when � � Hn o U�n � � is a lattice� that is thequotient Hn�� is compact� the existence of such �nite cover of Hn�� by a closednilpotent manifold Hn�� is due to Gromov �Gr and Buser�Karcher �BK resultsfor almost �at manifolds�Our proof of Theorem �� has the following scheme� Firstly� passing to a �nite

index subgroup� we may assume that the group � is torsion�free� After that� weshall �nd a �nite index subgroup �� whose rotational part is �good�� Then we

�� BORIS APANASOV

shall express the vector bundle Hn�� H�� as the Whitney sum of a trivialbundle and a �ber product� We �nish the proof by using the following criterionabout the triviality of �ber products�

Lemma �� Let F H V be a �ber product and suppose that the homomorphism� � H � GL�V � extends to a homomorphism � � F � GL�V �� Then F H V is atrivial bundle� F H V �� F�H V �

Proof� The isomorphism FHV �� F�HV is given by �f� v� �Hf� ��f��v��

�� Geometrical finiteness in complex hyperbolic geometry

Our main assumption on a complex hyperbolic n�manifold M is the geometrical�niteness of its fundamental group ��M� � G � PU�n� �� which in particularimplies that the discrete group G is �nitely generated�Here a subgroup G � PU�n� � is called discrete if it is a discrete subset of

PU�n� �� The limit set ��G� � �H nCof a discrete group G is the set of accumulation

points of �any� orbit G�y�� y � H nC� The complement of ��G� in �H n

Cis called the

discontinuity set !�G�� A discrete group G is called elementary if its limit set ��G�consists of at most two points� An in�nite discrete group G is called parabolic ifit has exactly one �xed point �x�G�" then ��G� � �x�G�� and G consists of eitherparabolic or elliptic elements� As it was observed by many authors �cf� �MaG��parabolicity in the variable curvature case is not as easy a condition to deal withas it is in the constant curvature space� However the results of x simplify thesituation� especially for geometrically �nite groups�Geometrical �niteness has been essentially used for real hyperbolic manifolds�

where geometric analysis and ideas of Thurston provided powerful tools for under�standing of their structure� Due to the absence of totally geodesic hypersurfacesin a space of variable negative curvature� we cannot use the original de�nition ofgeometrical �niteness which came from an assumption that the corresponding realhyperbolic manifold M � H n�G may be decomposed into a cell by cutting alonga �nite number of its totally geodesic hypersurfaces� that is the group G shouldpossess a �nite�sided fundamental polyhedron� see �Ah� However� we can de�negeometrically �nite groups G � PU�n� � as those ones whose limit sets ��G� consistof only conical limit points and parabolic �cusp� points p with compact quotients��G�nfpg��Gp with respect to parabolic stabilizers Gp � G of p� see �BM� Bow�There are other de�nitions of geometrical �niteness in terms of ends and the mini�mal convex retract of the noncompact manifoldM � which work well not only in thereal hyperbolic spaces H n �see �Mar� Th� A� A�� but also in spaces with variablepinched negative curvature �Bow�Our study of geometrical �niteness in complex hyperbolic geometry is based

on analysis of geometry and topology of thin �parabolic� ends of correspondingmanifolds and parabolic cusps of discrete isometry groups G � PU�n� ��Namely� suppose a point p � �H n

Cis �xed by some parabolic element of a given

discrete group G � PU�n� �� and Gp is the stabilizer of p in G� Conjugating G by

COMPLEX HYPERBOLIC AND CR�MANIFOLDS ��

an element hp � PU�n� � � hp�p� � �� we may assume that the stabilizer Gp is asubgroup G� � H�n�� In particular� if p is the origin � � Hn� the transformationh� can be taken as the Heisenberg inversion I in the hyperchain �H

n��C

� It preservesthe unit Heisenberg sphere Sc�� f� � v� � Hn � jj� � v�jjc � g and acts in Hn

as follows�

I� � v� �

�

j j� � iv�

�v

v� � j j

�where � � v� � Hn � C

n�� R � ��

For any other point p� we may take hp as the Heisenberg inversion Ip whichpreserves the unit Heisenberg sphere Sc�p� � � f� � v� � �c�p� � � v�� g centeredat p� The inversion Ip is conjugate of I by the Heisenberg translation Tp and mapsp to ��After such a conjugation� we can apply Theorem �� to the parabolic stabilizer

G� � H�n� and get a connected Lie subgroup H� � Hn preserved by G� �up tochanging the origin�� So we can make the following de�nition�

Denition �� A set Up�r � H nCnfpg is called a standard cusp neighborhood of

radius r � � at a parabolic �xed point p � �H nCof a discrete group G � PU�n� � if�

for the Heisenberg inversion Ip � PU�n� � with respect to the unit sphere Sc�p� ��Ip�p� �� the following conditions hold�

�� Up�r � I��p �fx � HnC�Hn � �c�x�H�� rg� "

� � Up�r is precisely invariant with respect to Gp � G� that is�

��Up�r� � Up�r for � � Gp and g�Up�r� � Up�r � � for g � GnGp �

A parabolic point p � �H nCof G � PU�n� � is called a cusp point if it has a cusp

neighborhood Up�r�

We remark that some parabolic points of a discrete group G � PU�n� � may notbe cusp points� see examples in x�� of �AX� Applying Theorem �� and �Bow�we have�

Lemma �� Let p � �H nCbe a parabolic �xed point of a discrete subgroup G in

PU�n� �� Then p is a cusp point if and only if ��G�nfpg��Gp is compact�

This and �niteness results of Bowditch �B allow us to use another equivalentde�nitions of geometrical �niteness� In particular it follows that a discrete subgroupG in PU�n� � is geometrically �nite if and only if its quotient space M�G� ��H n

C� !�G��G has �nitely many ends� and each of them is a cusp end� that is an

end whose neighborhood can be taken �for an appropriate r � �� in the form�

Up�r�Gp �Sp�r�Gp� ��

whereSp�r � �HUp�r � I��p �fx � Hn

C �Hn � �c�x�H�� rg� �

� BORIS APANASOV

Now we see that a geometrically �nite manifold can be decomposed into a com�pact submanifold and �nitely many cusp submanifolds of the form �� Clearly�each of such cusp ends is homotopy equivalent to a Heisenberg � n � ��manifoldand moreover� due to Theorem �� to a compact k�manifold� k � n�� From thelast fact� it follows that the fundamental group of a Heisenberg manifold is �nitelypresented� and we get the following �niteness result�

Corollary �� Geometrically �nite groups G � PU�n� � are �nitely presented�

In the case of variable curvature� it is problematic to use geometric methodsbased on consideration of �nite sided fundamental polyhedra� in particular� Dirich�let polyhedra Dy�G� for G � PU�n� � bounded by bisectors in a complicated way�see � Mo � GP� FG� In the case of discrete parabolic groups G � PU�n� �� onemay expect that the Dirichlet polyhedron Dy�G� centered at a point y lying in a G�invariant subspace has �nitely many sides� It is true for real hyperbolic spaces �Aas well as for cyclic and dihedral parabolic groups in complex hyperbolic spaces�Namely� due to � Ph� Dirichlet polyhedra Dy�G� are always two sided for any cyclicgroup G � PU�n� � generated by a Heisenberg translation� Due the main resultin �GP� this �niteness also holds for a cyclic ellipto�parabolic group or a dihedralparabolic group G � PU�n� � generated by inversions in asymptotic complex hy�perplanes in H

nCif the central point y lies in a G�invariant vertical line or R�plane

�for any other center y� Dy�G� has in�nitely many sides�� Our technique easilyimplies that this �niteness still holds for generic parabolic cyclic groups �AX�

Theorem �� For any discrete group G � PU�n� � generated by a parabolicelement� there exists a point y� � H

nCsuch that the Dirichlet polyhedron Dy��G�

centered at y� has two sides�

Proof� Conjugating G and due to Theorem �� we may assume that G preservesa one dimensional subspace H� � Hn as well as H� R� � H n

C� where G acts

by translations� So we can take any point y� � H� R� as the central point of�two�sided� Dirichlet polyhedron Dy��G� because its orbit G�y�� coincides with theorbit G��y�� of a cyclic group generated by the Heisenberg translation induced by G�

�

However� the behavior of Dirichlet polyhedra for parabolic groups G � PU�n� �of rank more than one can be very bad� It is given by our construction �AX�where we have evaluated intersections of Dirichlet bisectors with a �dimensionalslice�

Theorem �� Let G � PU� � � be a discrete parabolic group conjugate to thesubgroup � � f�m�n� � C R � m�n � Zg of the Heisenberg group H� � C R�Then any Dirichlet polyhedron Dy�G� centered at any point y � H �

Chas in�nitely

many sides�

Despite the above example� the below application of Theorem �� provides aconstruction of fundamental polyhedra P �G� � H n

Cfor arbitrary discrete parabolic


groups G � PU�n� �� which are bounded by �nitely many hypersurfaces �di�er�ent from Dirichlet bisectors�� This result may be seen as a base for extension ofApanasov�s construction �A of �nite sided pseudo�Dirichlet polyhedra in H n tothe case of the complex hyperbolic space H n

C�

Theorem �� For any discrete parabolic group G � PU�n� �� there exists a�nite�sided fundamental polyhedron P �G� � H n

C�

Proof� After conjugation� we may assume that G � Hn o U�n � �� Let H� �Hn � C n�� R be the connected G�invariant subgroup given by Theorem �� Fora �xed u� � �� we consider the horocycle Vu� � H� fu�g � C n�� R R� �HnC� For distinct points y� y� � Vu� � the bisector C�y� y

�� fz � HnC� d�z� y� �

d�z� y��g intersects Vu� transversally� Since Vu� is G�invariant� its intersection witha Dirichlet polyhedron

Dy�G� ��

g�Gnfidg

fw � HnC � d�w� y� � d�w� g�y��g

centered at a point y � Vu� is a fundamental polyhedron for the G�action on Vu� �The polyhedron Dy�G��Vu� is compact due to Theorem �� and hence has �nitelymany sides� Now� considering G�equivariant projections �AX�

� � Hn �H� � �� H nC � Hn R� � Vu� � ��x� u� � ��x�� u��

we get a �nite�sided fundamental polyhedron ��Dy�G� � Vu�� for the action ofG in H n

C�

�

Another important application of Theorem �� shows that cusp ends of a geo�metrically �nite complex hyperbolic orbifoldsM have� up to a �nite covering ofM �a very simple structure�

Theorem �� Let G � PU�n� � be a geometrically �nite discrete group� ThenG has a subgroup G� of �nite index such that every parabolic subgroup of G� isisomorphic to a discrete subgroup of the Heisenberg group Hn � C n�� R � Inparticular� each parabolic subgroup of G� is free Abelian or ��step nilpotent�

The proof of this fact �AX is based on the residual �niteness of geometrically�nite subgroups in PU�n� � and the following two lemmas�

Lemma �� Let G � HnoU�n�� be a discrete group and HG � Hn a minimalG�invariant connected Lie subgroup �given by Theorem �� Then G acts on HG

by translations if G is either Abelian or ��step nilpotent�

Lemma �� Let G � Hn o U�n� � be a torsion free discrete group� F a �nitegroup and � � G �� F an epimorphism� Then the rotational part of ker�� hasstrictly smaller order than that of G if one of the following happens

�� G contains a �nite index Abelian subgroup and F is not Abelian�� G contains a �nite index ��step nilpotent subgroup and F is not a ��step

nilpotent group�


We remark that the last Lemma generalizes a result of C�S�Aravinda and T�Farrell�AF for Euclidean crystallographic groups�We conclude this section by pointing out that the problem of geometrical �nite�

ness is very di�erent in complex dimension two� Namely� it is a well known fact thatany �nitely generated discrete subgroup of PU�� or PO� � � is geometrically�nite� This and Goldman�s � Go local rigidity theorem for cocompact latticesG � U�� PU� � � allow us to formulate the following conjecture�

Conjecture �� All �nitely generated discrete groups G � PU� � � with non�empty discontinuity set !�G� � �H �

Care geometrically �nite�

�� Complex homology cobordisms and the boundary at infinity

The aim of this section is to study the topology of complex analytic �Kleinian�manifolds M�G� � �H n

C� !�G��G with geometrically �nite holonomy groups G �

PU�n� �� The boundary of this manifold� �M � !�G��G� has a spherical CR�structure and� in general� is non�compact�We are especially interested in the case of complex analytic surfaces� where

powerful methods of ��dimensional topology may be used� It is still unknown whatare suitable cuts of ��manifolds� which �conjecturally� split them into geometricblocks �alike Jaco�Shalen�Johannson decomposition of ��manifolds in Thurston�sgeometrization program" for a classi�cation of ��dimensional geometries� see �F�Wa�� Nevertheless� studying of complex surfaces suggests that in this case onecan use integer homology ��spheres and �almost �at� ��manifolds �with virtuallynilpotent fundamental groups�� Actually� as Sections � and � show� the lattermanifolds appear at the ends of �nite volume complex hyperbolic manifolds� Asit was shown by C�T�C�Wall �Wa� the assignment of the appropriate ��geometry�when available� gives a detailed insight into the intrinsic structure of a complexsurface� To identify complex hyperbolic blocks in such a splitting� one can use Yau�suniformization theorem �Ya� It implies that every smooth complex projective �surface M with positive canonical bundle and satisfying the topological conditionthat ��M� � � Signature�M�� is a complex hyperbolic manifold� The necessityof homology sphere decomposition in dimension four is due to M�Freedman andL�Taylor result �� FT��

Let M be a simply connected �manifold with intersection form qM which de�composes as a direct sum qM � qM�

� qM�� where M��M� are smooth manifolds�

Then the manifold M can be represented as a connected sum M �M�#M� alonga homology sphere $�

Let us present an example of such a splitting�M � X#Y � of a simply connectedcomplex surface M with the intersection form QM into smooth manifolds �withboundary� X and Y � along a Z�homology ��sphere $ such that QM � QX � QY �Here one should mention that though X and Y are no longer closed manifolds� theintersection forms QX and QY are well de�ned on the second cohomology and areunimodular due to the condition that $ is a Z�homology ��sphere�


Example �� Let M be the Kummer surface

K� � f�z�� z�� z�� z� � C P� � z� � z� � z� � z� � �g �

Then there are four disjointly embedded �Seifert �bered� Z�homology �spheres inM � which split the Kummer surface into �ve blocks

K� � X� � Y� �� Y� �� Y� �� X� �

with intersection forms QXjand QYi equal E� and H� respectively

E� �

�BBBBBBBBB�

� � � � � � � � � � � � ��

�CCCCCCCCCA

� H �

��

��

Here the Z�homology spheres $ and $� are correspondingly the Poincar�e homologysphere $� � �� and Seifert �bered homology sphere $� � �� the minus sign meansthe change of orientation�

Scheme of splitting� Due to J�Milnor �Mil �see also �RV�� all Seifert �bered ho�mology ��spheres $ can be seen as the boundaries at in�nity of �geometrically�nite� complex hyperbolic orbifolds H �

C�� where the fundamental groups ��$� �

� � PU� � � act free in the sphere at in�nity �H �C� H�� In particular� the

Seifert �bered homology sphere $� � $� � �� is di�eomorphic to the quotient��C R�n�f�g R�� Here ��C R�n�f�g R� is the complement in the��sphere H� � �B�

Cto the boundary circle at in�nity of the complex geodesic

B�C� �C f�g�� and the group �� PU� � � acts on this complex geodesic

as the standard triangle group � � �� in the disk Poincar%e model of the hyperbolic �plane H �

R�

This homology ��sphere $� embeds in the K��surface M � splitting it into sub�manifolds with intersection forms E��H and E�� H� This embedding is describedin �Lo and �FS� One can keep decomposing the obtained two manifolds as in �FS and �nally split it into �ve pieces� Among additional embedded homology spheres�there is the only one known homology ��sphere with �nite fundamental group� thePoincar%e homology sphere $ � $� � �� One can introduce a spherical geometryon $ by representing ��$� as a �nite subgroup �� of the orthogonal groupO�� acting free on S� � �B�

C� Then $� � �� S�� can obtained by

identifying the opposite sides of the spherical dodecahedron whose dihedral anglesare �� see �KAG� �


However we note that it is unknown whether the obtained blocks may supportsome homogeneous ��geometries classi�ed by Filipkiewicz �F and �from the point

of view of Ka�hler structures� C�T�C�Wall �Wa� This raises a question whetherhomogeneous geometries or splitting along homology spheres �important from thetopological point of view� are relevant for a geometrization of smooth ��dimensionalmanifolds� For example� neither of Yi blocks in Example �� with the intersectionform H� can support a complex hyperbolic structure �which is a natural geometriccandidate since $ has a spherical CR�structure� because each of them has twocompact boundary components�In fact� in a sharp contrast to the real hyperbolic case� for a compact manifold

M�G� �that is for a geometrically �nite group G � PU�n� � without cusps�� anapplication of Kohn�Rossi analytic extension theorem shows that the boundary ofM�G� is connected� and the limit set ��G� is in some sense small �see � EMM and�for quaternionic and Caley hyperbolic manifolds � C� CI�� Moreover� according toa recent result of D�Burns �see also Theorem �� in �NR�� the same claim aboutconnectedness of the boundary �M�G� still holds if only a boundary componentis compact� �In dimension n � �� D�Burns theorem based on �BuM uses the lastcompactness condition to prove geometrical �niteness of the whole manifoldM�G��see also �NR ��However� if no component of �M�G� is compact and we have no �niteness condi�

tion on the holonomy group of the complex hyperbolic manifoldM�G�� the situationis completely di�erent due to our construction �AX�

Theorem �� In any dimension n � and for any integers k� k�� k � k� � ��there exists a complex hyperbolic n�manifold M � H

nC�G� G � PU�n� �� whose

boundary at in�nity splits up into k connected manifolds� ��M � N� � � � � � Nk�Moreover� for each boundary component Nj� j � k�� its inclusion into the manifoldM�G�� ij � Nj �M�G�� induces a homotopy equivalence of Nj to M�G��

For a torsion free discrete group G � PU�n� �� a connected component !� ofthe discontinuity set !�G� � �H n

Cwith the stabilizer G� � G is contractible and

G�invariant if and only if the inclusion N� � !��G� � M�G� induces a homotopyequivalence of N� to M�G� �A� AX� It allows us to reformulate Theorem �� as

Theorem �� In any complex dimension n � and for any natural numbers kand k�� k � k� � �� there exists a discrete group G � G�n� k� k�� PU�n� �whose discontinuity set !�G� � �H n

Csplits up into k G�invariant components�

!�G� � !� � � � � � !k� and the �rst k� components are contractible�

Sketch of Proof� To prove this claim �see �AX for details�� it is crucial to constructa discrete group G � PU�n� � whose discontinuity set consists of two G�invarianttopological balls� To do that� we construct an in�nite family $ of disjoint closedHeisenberg balls Bi � B�ai� ri� � �H n

Csuch that the complement of their clo�

sure� �H nCnSiB�ai� ri� � P� � P�� consists of two topological balls� P� and P��

In our construction of such a family $ of H�balls Bj � we essentially relie on thecontact structure of the Heisenberg group Hn� Namely� $ is the disjoint union of


�nite sets $i of closed H�balls whose boundary H�spheres have �real hyperspheres�serving as the boundaries of � n � ��dimensional cobordisms Ni� In the limit�these cobordisms converge to the set of limit vertices of the polyhedra P� and P�which are bounded by the H�spheres Sj � �Bj� Bj � $� Then the desired groupG � G�n� � � � PU�n� � is generated by involutions Ij which preserve those real� n� ��spheres lying in Sj � �P� � �P�� see Fig��

Figure �� Cobordism N� in H with two boundary real circles

We notice that� due to our construction� the intersection of each H�sphere Sjand each of the polyhedra P� and P� in the complement to the balls Bj � $ is atopological � n� ��ball which splits into two sides� Aj and A

�j � and Ii�Ai� � A�i�

This allows us to de�ne our desired discrete group G � G � PU�n� � as thediscrete free product� G � �j �j � �i hIji� of in�nitely many cyclic groups �jgenerated by involutions Ij with respect to the H�spheres Sj � �Bj� So P� � P�is a fundamental polyhedron for the action of G in �H n

C� and sides of each of its

connected components� P� or P�� are topological balls pairwise equivalent withrespect to the corresponding generators Ij � G� Applying standard arguments �see�A� Lemmas �� &�� we see that the discontinuity set !�G� � Hn consists of

two G�invariant topological balls !� and !�� !k � int�S

g�G g�Pk� � k � � � The

fact that !k is a topological ball follows from the observation that this domain isthe union of a monotone sequence�

V� � int�Pk� � V� � intPk � I��Pk�

� V� � � � � �


of open topological balls� see �Br� Note that here we use the property of ourconstruction that Vi is always a topological ball�In the general case of k � k� � �� k � �� we can apply the above in�nite free prod�

ucts and our cobordism construction of in�nite families of H�balls with preassignedproperties in order to �su�ciently closely� �approximate� a given hypersurface inHn by the limit sets of constructed discrete groups� For such hypersurfaces� we usethe so called �tree�like surfaces� which are boundaries of regular neighborhoods oftrees inHn� This allows us to generalize A�Tetenov�s �T� KAG construction of dis�crete groups G on the m�dimensional sphere Sm � m � �� whose discontinuity setssplit into any given number k of G�invariant contractible connected components�

�

Although� in the general case of complex hyperbolic manifolds M with �nitelygenerated ��M� �� G� the problem on the number of boundary components ofM�G� is still unclear� we show below that the situation described in Theorem ��is impossible if M is geometrically �nite� We refer the reader to �AX for moreprecise formulation and proof of this cobordism theorem�

Theorem �� Let G � PU�n� � be a geometrically �nite non�elementary tor�sion free discrete group whose Kleinian manifold M�G� has non�compact boundary�M � !�G��G with a component N� � �M homotopy equivalent to M�G�� Thenthere exists a compact homology cobordism Mc �M�G� such that M�G� can be re�constructed from Mc by gluing up a �nite number of open collars Mi �� whereeach Mi is �nitely covered by the product EkB�n�k�� of a closed ��n��k��ball anda closed k�manifold Ek which is either �at or a nil�manifold �with ��step nilpotentfundamental group��

In connection to this cobordism theorem� it is worth to mention another inter�esting fact due to Livingston�Myers �My construction� Namely� any Z�homology ��sphere is homology cobordant to a real hyperbolic one� However� it is still unknownwhether one can introduce a geometric structure on such a homology cobordism�or a CR�structure on a given real hyperbolic ��manifold �in particular� a homologysphere� or on a Z�homology ��sphere of plumbing type� We refer to �S� Mat forrecent advances on homology cobordisms� in particular� for results on Floer homol�ogy of homology ��spheres and a new Saveliev�s �presumably� homology cobordism�invariant based on Floer homology�

�� Homeomorphisms induced by group isomorphisms

As another application of the developed methods� we study the following wellknown problem of geometric realizations of group isomorphisms�

Problem �� Given a type preserving isomorphism � � G� H of discrete groupsG�H � PU�n� �� nd subsets XG� XH � H n

Cinvariant for the action of groups G

and H� respectively� and an equivariant homeomorphism f� � XG � XH which in�duces the isomorphism �� Determine metric properties of f�� in particular� whetherit is either quasisymmetric or quasiconformal�


Such type problems were studied by several authors� In the case of lattices Gand H in rank symmetric spaces X� G�Mostow �Mo proved in his celebratedrigidity theorem that such isomorphisms � � G � H can be extended to innerisomorphisms of X� provided that there is no analytic homomorphism of X ontoPSL� �R�� For that proof� it was essential to prove that � can be induced by aquasiconformal homeomorphism of the sphere at in�nity �X which is the one pointcompacti�cation of a �nilpotent� Carnot group N �for quasiconformal mappings inHeisenberg and Carnot groups� see �KR� P��If geometrically �nite groups G�H � PU�n� � have parabolic elements and

are neither lattices nor trivial� the only results on geometric realization of theirisomorphisms are known in the real hyperbolic space �Tu� Generally� those methodscannot be used in the complex hyperbolic space due to lack of control over convexhulls �where the convex hull of three points may be ��dimensional�� especially nearbycusps� Another �dynamical� approach due to C�Yue �Yu � Cor�B �and the Anosov�Smale stability theorem for hyperbolic �ows� can be used only for convex cocompactgroups G and H� see �Yu�� As a �rst step in solving the general Problem �� wehave the following isomorphism theorem �A��

Theorem �� Let � � G � H be a type preserving isomorphism of two non�ele�mentary geometrically �nite groups G�H � PU�n� �� Then there exists a uniqueequivariant homeomorphism f� � ��G�� H� of their limit sets that induces theisomorphism �� Moreover� if ��G� � �H n

C� the homeomorphism f� is the restriction

of a hyperbolic isometry h � PU�n� ��

Proof� To prove this claim� we consider the Cayley graphK�G� �� of a group G witha given �nite set � of generators� This is a �complex whose vertices are elements ofG� and such that two vertices a� b � G are joined by an edge if and only if a � bg��

for some generator g � �� Let j � j be the word norm on K�G� �� that is� jgj equalsthe minimal length of words in the alphabet � representing a given element g � G�Choosing a function � such that ��r� � �r� for r � � and �� one can de�nethe length of an edge �a� b � K�G� �� as d��a� b� � minf��jaj�� jbj�g� Consideringpaths of minimal length in the sense of the function d��a� b�� one can extend it to

a metric on the Cayley graph K�G� �� So taking the Cauchy completion K�G� ��of that metric space� we have the de�nition of the group completion G as thecompact metric space K�G� ��nK�G� �� see �Fl� Up to a Lipschitz equivalence�this de�nition does not depend on �� It is also clear that� for a cyclic group Z� itscompletion Z consists of two points� Nevertheless� for a nilpotent group G with oneend� its completion G is a one�point set �Fl�Now we can de�ne a proper equivariant embedding F � K�G� �� H n

Cof the

Cayley graph of a given geometrically �nite group G � PU�n� �� To do that wemay assume that the stabilizer of a point� say � � H n

C� is trivial� Then we set

F �g� � g�� for any vertex g � K�G� �� and F maps any edge �a� b � K�G� �� tothe geodesic segment �a�� b�� H n

C�

Proposition �� For a geometrically �nite discrete group G � PU�n� �� there

� BORIS APANASOV

are constants K�K � � � such that the following bounds hold for all elements g � Gwith jgj � K �

ln� jgj �K�� lnK� � d�� g�� Kjgj � ��

The proof of this claim is based on a comparison of the Bergman metric d�� and the path metric d�� on the following subset bh� � H

nC� Let C��G�� H

nC

be the convex hull of the limit set ��G� � �H nC� that is the minimal convex subset in

H nCwhose closure in H n

Ccontains ��G�� Clearly� it is G�invariant� and its quotient

C��G��G is the minimal convex retract of H nC�G� Since G is geometrically �nite�

the complement inM�G� to neighbourhoods of ��nitely many� cusp ends is compactand correspond to a compact subset in the minimal convex retract� which canbe taken as H ��G� In other words� H � � C��G�� is the complement in theconvex hull to a G�invariant family of disjoint horoballs each of which is strictlyinvariant with respect to its �parabolic� stabilizer in G� see �AX� Bow� cf� also�A� Th� �� Now� having co�compact action of the group G on the domainH � whose boundary includes some horospheres� we can reduce our comparison ofdistances d � d�x� x�� and d� � d��x� x

�� to their comparison on a horosphere� Sowe can take points x � �� u� and x� � � � v� u� on a �horizontal� horosphereSu � C n�� R fug � H n

C� Then the distances d and d� are as follows �Pr �

cosh�d

�

�u�j j � �uj j� � �u� � v�

� d��

j j�

u�

v�

�u��

This comparison and the basic fact due to Cannon �Can that� for a co�compactaction of a group G in a metric space X� its Cayley graph can be quasi�isometricallyembedded into X� �nish our proof of ��Now we apply Proposition �� to de�ne a G�equivariant extension of the map

F from the Cayley graph K�G� �� to the group completion G� Since the groupcompletion of any parabolic subgroup Gp � G is either a point or a two�point set�depending on whether Gp is a �nite extension of cyclic or a nilpotent group withone end�� we get

Theorem �� For a geometrically �nite discrete group G � PU�n� �� there is acontinuous G�equivariant map G � G� ��G�� Moreover� the map G is bijectiveeverywhere but the set of parabolic �xed points p � ��G� whose stabilizers Gp � Ghave rank one� On this set� the map G is two�to�one�

Now we can �nish our proof of Theorem �� by looking at the following diagramof maps�

��G��G�� G

�� H

�H�� H� �

where the homeomorphism � is induced by the isomorphism �� and the continuousmaps G and H are de�ned by Theorem �� Namely� one can de�ne a mapf� � H�

��G � Here the map ��G is the right inverse to G� which exists due

to Theorem �� Furthermore� the map ��G is bijective everywhere but the set ofparabolic �xed points p � ��G� whose stabilizers Gp � G have rank one� where


it is �to�� Hence the composition map f� is bijective and G�equivariant� Itsuniqueness follows from its continuity and the fact that the image of the attractive�xed point of an loxodromic element g � G must be the attractive �xed point ofthe loxodromic element ��g� � H �such loxodromic �xed points are dense in thelimit set� see �A��The last claim of the Theorem �� directly follows from the Mostow rigidity

theorem �Mo because a geometrically �nite group G � PU�n� � with ��G� � �H nC

is co��nite� vol �H nC�G� ��

�

Remark �� Our proof of Theorem �� can be easily extended to the general sit�uation� that is� to construct equivariant homeomorphisms f� � ��G� � ��H�conjugating the actions �on the limit sets� of isomorphic geometrically �nite groupsG�H � IsomX in a �symmetric� space X with pinched negative curvature K��b� � K � �a� � �� Actually� bounds similar to �� in Prop� �� crucial for ourargument� can be obtained from a result due to Heintze and Im Hof �HI� Th��which compares the geometry of horospheres Su � X with that in the spaces ofconstant curvature �a� and �b�� respectively� It gives� that for all x� y � Su andtheir distances d � d�x� y� and du � du�x� y� in the space X and in the horosphereSu� respectively� one has that

�asinh�a � d� � � du �

�bsinh�b � d� ��

Upon existence of such homeomorphisms f� inducing given isomorphisms � ofdiscrete subgroups of PU�n� �� the Problem �� can be reduced to the questionswhether f� 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 thesphere at in�nity �X or even to the whole space H n

C� inducing the isomorphism ��

For convex cocompact groups obtained by nearby representations� this may be seenas a generalization of D�Sullivan stability theorem �Su � see also �A��However� in a deep contrast to the real hyperbolic case� here we have an interest�

ing e�ect related to possible noncompactness of the boundary �M�G� � !�G��G�Namely� even for the simplest case of parabolic cyclic groupsG �� H � PU�n� �� thehomeomorphic CR�manifolds �M�G� � Hn�G and �M�H� � Hn�H may be notquasiconformally equivalent� see �Min� In fact� among such Cauchy�Riemannian ��manifolds �homeomorphic to R�S�� there are exactly two quasiconformal equiva�lence classes whose representatives have the holonomy groups generated correspond�ingly by a vertical H�translation by �� C R and a horizontal H�translationby �� C R�Theorem �� presents a more sophisticated topological deformation ff�g� f� �

H �C� H �

C� of a �complex�Fuchsian� co��nite group G � PU�� PU� � � to

quasi�Fuchsian discrete groups G� � f�Gf�� PU� � �� It deforms pure para�

bolic subgroups in G to subgroups in G� generated by Heisenberg �screw transla�tions�� As we point out� any such G�equivariant conjugations of the groups G andG� cannot be contactomorphisms because they must map some poli of Dirichletbisectors to non�poli ones in the image�bisectors" moreover� they cannot be qua�siconformal� either� This shows the impossibility of the mentioned extension of

BORIS APANASOV

Sullivan�s stability theorem to the case of groups with rank one cusps�Also we note that� besides the metrical �quasisymmetric� part of the geometriza�

tion Problem �� there are some topological obstructions for extensions of equivari�ant homeomorphisms f�� f� � ��G�� H�� It follows from the next example�

Example �� Let G � PU�� PU� � � and H � PO� � � � PU� � � betwo geometrically �nite �loxodromic� groups isomorphic to the fundamental group��Sg� of a compact oriented surface Sg of genus g � � Then the equivarianthomeomorphism f� � ��G� � ��H� cannot be homeomorphically extended to thewhole sphere �H �

C S��

Proof� The obstruction in this example is topological and is due to the fact that thequotient manifolds M� � H �

C�G and M� � H �

C�H are not homeomorphic� Namely�

these complex surfaces are disk bundles over the Riemann surface Sg and havedi�erent Toledo invariants� ��H �

C�G� � g � and ��H �

C�H� � �� see �To�

The complex structures of the complex surfaces M� and M� are quite di�erent�too� The �rst manifold M� has a natural embedding of the Riemann surface Sgas a holomorphic totally geodesic closed submanifold� and hence M� cannot be aStein manifolds� The second manifolds M� is a Stein manifold due to a result byBurns�Shnider �BS� Moreover due to Goldman �Go� since the surface Sp � M�

is closed� the manifold M� is locally rigid in the sense that every nearby represen�tation G � PU� � � stabilizes a complex geodesic in H �

Cand is conjugate to a

representation G � PU�� PU� � �� In other words� there are no non�trivial�quasi�Fuchsian� deformations of G and M�� On the other hand� as we show inthe next section �cf� Theorem �� the second manifold M� has plentiful enoughTeichm�uller space of di�erent �quasi�Fuchsian� complex hyperbolic structures�

�

�� Deformations of complex hyperbolic and

CR�structures flexibility versus rigidity

Since any real hyperbolic n�manifold can be �totally geodesically� embedded to acomplex hyperbolic n�manifold H n

C�G� �exibility of the latter ones is evident start�

ing with hyperbolic structures on a Riemann surface of genus g � � which formTeichm�uller space� a complex analytic ��g � ��manifold� Strong rigidity startsin real dimension �� Namely� due to the Mostow rigidity theorem �M� hyper�bolic structures of �nite volume and �real� dimension at least three are uniquelydetermined by their topology� and one has no continuous deformations of them�Yet hyperbolic ��manifolds have plentiful enough in�nitesimal deformations and�according to Thurston�s hyperbolic Dehn surgery theorem �Th� noncompact hy�perbolic ��manifolds of �nite volume can be approximated by compact hyperbolic��manifolds�Also� despite their hyperbolic rigidity� real hyperbolic manifolds M can be de�

formed as conformal manifolds� or equivalently as higher�dimensional hyperbolicmanifoldsM�� of in�nite volume� First such quasi�Fuchsian deformations weregiven by the author �A and� after Thurston�s �Mickey Mouse� example �Th� they


were called bendings ofM along its totally geodesic hypersurfaces� see also �A� A �A��A�� JM� Ko� Su� Furthermore� all these deformations are quasiconformallyequivalent showing a rich supply of quasiconformal G�equivariant homeomorphismsin the real hyperbolic space H n

R� In particular� the limit set ��G� � �H n��

Rdeforms

continuously from a round sphere �H nR� Sn�� Sn � H

n��R

into nondi�erentiablyembedded topological �n� ��spheres quasiconformally equivalent to Sn��Contrasting to the above �exibility� �non�real� hyperbolic manifolds seem much

more rigid� In particular� due to Pansu �P� quasiconformal maps in the sphere atin�nity of quaternionic�octionic hyperbolic spaces are necessarily automorphisms�and thus there cannot be interesting quasiconformal deformations of correspondingstructures� Secondly� due to Corlette�s rigidity theorem �Co � such manifolds areeven super�rigid � analogously to Margulis super�rigidity in higher rank �MG�Furthermore� complex hyperbolic manifolds share the above rigidity of quater�nionic�octionic hyperbolic manifolds� Namely� due to the Goldman�s local rigiditytheorem in dimension n � �G and its extension for n � � �GM� every nearbydiscrete representation � � G� PU�n� � of a cocompact lattice G � PU�n� � �stabilizes a complex totally geodesic subspace H

n��C

in H nC� Thus the limit set

��G� � �H nCis always a round sphere S�n�� In higher dimensions n � �� this

local rigidity of complex hyperbolic n�manifolds M homotopy equivalent to theirclosed complex totally geodesic hypersurfaces is even global due to a recent Yue�srigidity theorem �Yu�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 Mare not rigid in general �we suspect that it is true for all Stein manifolds with �big�ends at in�nity�� Such a �exibility has two aspects�First� we point out that the condition that the group G � PU�n� � preserves a

complex totally geodesic hyperspace in H nCis essential for local rigidity of deforma�

tions only for co�compact lattices G � PU�n � � �� This is due to the followingour result �ACG�

Theorem �� Let G � PU�� be a co��nite free lattice whose action in H�C

is generated by four real involutions �with �xed mutually tangent real circles atin�nity�� Then there is a continuous family ff�g� �� of G�equivariant

homeomorphisms in H �Cwhich induce non�trivial quasi�Fuchsian �discrete faithful�

representations f�� G � PU� � �� Moreover� for each � �� any G�equivariant

homeomorphism of H �Cthat induces the representation f�� cannot be quasiconformal�

This and an Yue�s �Yu result on Hausdor� dimension show that there aredeformations of a co��nite Fuchsian group G � PU�� into quasi�Fuchsian groupsG� � f�Gf

�� PU� � � with Hausdor� dimension of the limit set ��G�� strictly

bigger than one�Secondly� we point out that the noncompactness condition in the above non�

rigidity is not essential� either� Namely� complex hyperbolic Stein manifolds Mhomotopy equivalent to their closed totally real geodesic surfaces are not rigid� too�Namely� we give a canonical construction of continuous non�trivial quasi�Fuchsian

� BORIS APANASOV

deformations of manifolds M � dimC M � � �bered over closed Riemann surfaces�which are the �rst such deformations of �brations over compact base �for a non�compact base corresponding to an ideal triangle group G � PO� � �� see �GP ��Our construction is inspired by the approach the author used for bending defor�

mations of real hyperbolic �conformal� manifolds along totally geodesic hypersur�faces ��A � A�� and by an example of M�Carneiro�N�Gusevskii �Gu constructing anon�trivial discrete representation of a surface group into PU� � �� In the case ofcomplex hyperbolic �and Cauchy�Riemannian� structures� the constructed �bend�ings� work however in a di�erent way than in the real case� Namely our complexbending deformations involve simultaneous bending of the base of the �bration ofthe complex surface M as well as bendings of each of its totally geodesic �bers�see Remark �� Such bending deformations of complex surfaces are associatedto their real simple closed geodesics �of real codimension �� but have nothingcommon with the so called cone deformations of real hyperbolic ��manifolds alongclosed geodesics� see �A�� A��Furthermore� there are well known complications in constructing equivariant

homeomorphisms in the complex hyperbolic space and in Cauchy�Riemannian ge�ometry� which are due to necessary invariantness of the K�ahler and contact struc�tures �correspondingly in H n

Cand at its in�nity� Hn�� Despite that� the constructed

complex bending deformations are induced by equivariant homeomorphisms of H nC�

which are in addition quasiconformal�

Theorem �� Let G � PO� � � � PU� � � be a given �non�elementary� discretegroup� Then� for any simple closed geodesic � in the Riemann ��surface S � H�

R�G

and a su�ciently small � � �� there is a holomorphic family of G�equivariant

quasiconformal homeomorphisms F� � H �C� H �

C� �� which de�nes

the bending �quasi�Fuchsian� deformation B� � �� R��G� of the group Galong the geodesic �� B�� F �

� �

We notice that deformations of a complex hyperbolic manifold M may dependon many parameters described by the Teichm�uller space T �M� of isotopy classes ofcomplex hyperbolic structures onM � One can reduce the study of this space T �M�to studying the variety T �G� of conjugacy classes of discrete faithful representations� � G � PU�n� � �involving the space D�M� of the developing maps� see �Go �FG�� Here T �G� � R��G��PU�n� �� and the variety R��G� � Hom�G�PU�n� ��consists of discrete faithful representations � of the group G with in�nite co�volume�Vol�H n

C�G� �� In particular� our complex bending deformations depend on many

independent parameters as it can be shown by applying our construction and %ElieCartan �Car angular invariant in Cauchy�Riemannian geometry�

Corollary �� Let Sp � H �R�G be a closed totally real geodesic surface of genus

p � in a given complex hyperbolic surface M � H �C�G� G � PO� � � � PU� � ��

Then there is an embedding � � B � B�p�� T �M� of a real ��p � ��ball intothe Teichm�uller space of M � de�ned by bending deformations along disjoint closedgeodesics in M and by the projection � � D�M�� T �M� � D�M��PU� � � in the


development space D�M��

Basic Construction Proof of Theorem �� Now we start with a totallyreal geodesic surface S � H �

R�G in the complex surface M � H �

C�G� where G �

PO� � � � PU� � � is a given discrete group� and �x a simple closed geodesic �on S� We may assume that the loop � is covered by a geodesic A � H �

R� H �

C

whose ends at in�nity are � and the origin of the Heisenberg group H � C R�H � �H �

C� Furthermore� using quasiconformal deformations of the Riemann surface

S �in the Teichm�uller space T �S�� that is� by deforming the inclusion G � PO� � �in PO� � � by bendings along the loop �� see Corollary �� in �A�� we can assumethat the hyperbolic length of � is su�ciently small and the radius of its tubularneighborhood is big enough�

Lemma �� Let g� be a hyperbolic element of a non�elementary discrete groupG � PO� � � � PU� � � with translation length � along its axis A � H �

R� Then

any tubular neighborhood U�A� of the axis A of radius � � � is precisely invariantwith respect to its stabilizer G� � G if sinh�� sinh�� Furthermore�for su�ciently small �� the Dirichlet polyhedron Dz�G� � H �

Cof the group

G centered at a point z � A has two sides a and a� intersecting the axis A and suchthat g��a� � a��

Then the group G and its subgroups G�� G�� G� in the free amalgamated �orHNN�extension� decomposition of G have Dirichlet polyhedra Dz�Gi� � H

�C�

i � �� centered at a point z � A � �� whose intersections with thehyperbolic �plane H �

Rhave the shapes indicated in Figures ��

Figure �� G� � G � G� �G�G� Figure �� G� � G � G� �G�

G�

In particular we have that� except two bisectors S and S� that are equivalentunder the hyperbolic translation g� �which generates the stabilizer G� � G of theaxis A�� all other bisectors bounding such a Dirichlet polyhedron lie in su�cientlysmall �cone neighborhoods� C� and C� of the arcs �in�nite rays� R� and R� ofthe real circle R f�g � C R � H�Actually� we may assume that the Heisenberg spheres at in�nity of the bisectors

S and S� have radii and r� � � correspondingly� Then� for a su�ciently small ��

� � � �� r�� the cone neighborhoods C�� C� � H �Cnf�g � C R �� are

� BORIS APANASOV

Figure �� G� � G � G��G�Figure �� G � G��G�

correspondingly the cones of the ��neighborhoods of the points ��

C R �� with respect to the Cygan metric �c in H �Cnf�g� see � ��

Clearly� we may consider the length � of the geodesic � so small that closures

of all equidistant halfspaces in H �Cnf�g bounded by those bisectors �and whose

interiors are disjoint from the Dirichlet polyhedron Dz�G�� do not intersect theco�vertical bisector whose in�nity is iR R � C R� It follows from the fact �Go��Thm VII�� that equidistant half�spaces S� and S� in H �

Care disjoint if and

only if the half�planes S� � H �Rand S� � H �

Rare disjoint� see Figures ��

Now we are ready to de�ne a quasiconformal bending deformation of the groupG along the geodesic A� which de�nes a bending deformation of the complex surfaceM � H �

C�G along the given closed geodesic � � S �M �

We specify numbers and � such that � � � � �� and theintersection C� � �C f�g� is contained in the angle fz � C � j arg zj � �g� Thenwe de�ne a bending homeomorphism � � �� C � C which bends the real axisR � C at the origin by the angle � see Fig� ��

��z� �

��

z if j arg zj � � � �

z � exp�i� if j arg zj � �

z � exp�i�� arg z � �� if � � arg z � � � �

z � exp�i� � �arg z � �� if � � � � arg z � ��

��

Figure �


For negative � � � � � � �� we set ��z� � ��z�� Clearly� �� isquasiconformal with respect to the Cygan norm � �� and is an isometry in the��cone neighborhood of the real axis R because its linear distortion is given by

K�� z� �

��

if j arg zj � � � �

if j arg zj � �

�� if � � arg z � � � �

�� if � � � � arg z � ��

��

Foliating the punctured Heisenberg group Hnf�g by Heisenberg spheres S�� r�of radii r � �� we can extend the bending homeomorphism �� to an elementarybending homeomorphism � � �� H � H� �� of the wholesphere S� � H at in�nity�Namely� for the �dihedral angles� W��W� � H with the common vertical axis

f�g R and which are foliated by arcs of real circles connecting points �� v� and��v� on the vertical axis and intersecting the the ��cone neighborhoods of in�niterays R� �R� � C � correspondingly� the restrictions �jW�

and �jW�of the bending

homeomorphism � � �� are correspondingly the identity and the unitary rotationU� � PU� � � by angle about the vertical axis f�g R � H� see also �A�� Then it follows from �� that �� is a G��equivariant quasiconformal

homeomorphism in H�We can naturally extend the foliation of the punctured Heisenberg group Hnf�g

by Heisenberg spheres S�� r� to a foliation of the hyperbolic space H �Cby bisectors

Sr having those S�� r� as the spheres at in�nity� It is well known �see �M � thateach bisector Sr contains a geodesic �r which connects points ��r�� and �� r��of the Heisenberg group H at in�nity� and furthermore Sr �bers over �r by complexgeodesics Y whose circles at in�nity are complex circles foliating the sphere S�� r��Using those foliations of the hyperbolic space H �

Cand bisectors Sr� we extend

the elementary bending homeomorphism �� H � H at in�nity to an elemen�

tary bending homeomorphism �� H �C� H �

C� Namely� the map �� preserves

each of bisectors Sr� each complex geodesic �ber Y in such bisectors� and �xesthe intersection points y of those complex geodesic �bers and the complex geodesicconnecting the origin and � of the Heisenberg group H at in�nity� We completeour extension �� by de�ning its restriction to a given �invariant� complex geodesic�ber Y with the �xed point y � Y � This map is obtained by radiating the circlehomeomorphism �� j�Y to the whole �Poincar%e� hyperbolic �plane Y along geo�desic rays �y�� Y � so that it preserves circles in Y centered at y and bends �at y�by the angle � the geodesic in Y connecting the central points of the correspondingarcs of the complex circle �Y � see Fig��Due to the construction� the elementary bending �quasiconformal� homeomor�

phism �� commutes with elements of the cyclic loxodromic group G� � G� An�other most important property of the homeomorphism �� is the following�Let Dz�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� � G� and S� � H �C

� BORIS APANASOV

be a �half�space� disjoint from Dz�G� and bounded by a bisector S � H�Cwhich is

di�erent from bisectors Sr� r � �� and contains a side s of the polyhedron Dz�G��

Then there is an open neighborhood U�S�� H �Csuch that the restriction of the

elementary bending homeomorphism �� to it either is the identity or coincideswith the unitary rotation U� � PU� � � by the angle about the �vertical� complexgeodesic �containing the vertical axis f�g R � H at in�nity��The above properties of quasiconformal homeomorphism � � �� show that the

imageD� � ��Dz�G�� is a polyhedron in H�Cbounded by bisectors� Furthermore�

there is a natural identi�cation of its sides induced by �� Namely� the pairs ofsides preserved by � are identi�ed by the original generators of the group G� � G�For other sides s� of D�� which are images of corresponding sides s � Dz�G� underthe unitary rotation U�� we de�ne side pairings by using the group G decomposition�see Fig� ��Actually� if G � G� �G�

G�� we change the original side pairings g � G� ofDz�G��sides to the hyperbolic isometries U�gU

�� PU� � �� In the case of HNN�

extension� G � G��G�� hG�� g�i� we change the original side pairing g� � G of

Dz�G��sides to the hyperbolic isometry U�g� � PU� � �� In other words� we de�nedeformed groups G� � PU� � � correspondingly as

G� � G� �G�U�G�U

�� or G� � hG�� U�g�i � G� �G�

� ��

This shows that the family of representations G� G� � PU� � � does not dependon angles � and holomorphically depends on the angle parameter � Let us alsoobserve that� for small enough angles � the behavior of neighboring polyhedrag��D�� g

� � G� is the same as of those g�Dz�G�� g � G� around the Dirichletfundamental polyhedron Dz�G�� This is because the new polyhedron D� � H �

Chas

isometrically the same �tesselations of� neighborhoods of its side�intersections asDz�G� had� This implies that the polyhedra g

��D�� g� � G�� form a tesselation of

H �C�with non�overlapping interiors�� Hence the deformed group G� � PU� � � is

a discrete group� and D� is its fundamental polyhedron bounded by bisectors�Using G�compatibility of the restriction of the elementary bending homeomor�

phism � � �� to the closure Dz�G� � H �C� we equivariantly extend it from the

polyhedron Dz�G� to the whole space H �C� !�G� accordingly to the G�action�

In fact� in terms of the natural isomorphism � � G � G� which is identical onthe subgroup G� � G� we can write the obtained G�equivariant homeomorphism

F � F� � H �Cn��G�� H �

Cn��G�� in the following form�

F��x� � ��x� for x � Dz�G��

F� � g�x� � g� � F��x� for x � H �Cn��G�� g � G� g� � ��g� � G� �

��&�

Due to quasiconformality of �� the extended G�equivariant homeomorphismF� is quasiconformal� Furthermore� its extension by continuity to the limit �real�circle ��G� coincides with the canonical equivariant homeomorphism f� � ��G��


��G�� given by the isomorphism Theorem �� Hence we have a G�equivariant

quasiconformal self�homeomorphism of the whole space H �C� which we denote as

before by F��The family of G�equivariant quasiconformal homeomorphisms F� induces repre�

sentations F �� G � G� � F�G�F

�� In other words� we have

a curve B � �� R��G� in the variety R��G� of faithful discrete repre�sentations of G into PU� � �� which covers a nontrivial curve in the Teichm�ullerspace T �G� represented by conjugacy classes �B�� F �

� � We call the constructeddeformation B the bending deformation of a given lattice G � PO� � � � PU� � �along a bending geodesic A � H �

Cwith loxodromic stabilizer G� � G� In terms of

manifolds� B is the bending deformation of a given complex surface M � H �C�G

homotopy equivalent to its totally real geodesic surface Sg � M � along a givensimple geodesic ��

�

Remark �� It follows from the above construction of the bending homeomorphismF�� that the deformed complex hyperbolic surface M� � H

�C�G� �bers over the

pleated hyperbolic surface S� � F��H�R��G� �with the closed geodesic � as the

singular locus�� The �bers of this �bration are �singular real planes� obtainedfrom totally real geodesic �planes by bending them by angle along completereal geodesics� These �singular� real geodesics are the intersections of the complexgeodesic connecting the axis A of the cyclic group G� � G and the totally realgeodesic planes that represent �bers of the original �bration in M � H

�C�G�

Proof of Corollary �� Since� due to �� bendings along disjoint closed geodesicsare independent� we need to show that our bending deformation is not trivial� and�B�� B�� for any �� The non�triviality of our deformation follows directly from �� cf� �A�� Namely�

the restrictions ��jG�of bending representations to a non�elementary subgroup

G� � G �in general� to a �real� subgroup Gr � G corresponding to a totally realgeodesic piece in the homotopy equivalent surface S w M� are identical� So if thedeformation B were trivial then it would be conjugation of the group G by projec�tive transformations that commute with the non�trivial real subgroup Gr � G andpointwise �x the totally real geodesic plane H �

R� This contradicts to the fact that

the limit set of any deformed group G�� does not belong to the real circlecontaining the limit Cantor set ��Gr��

The injectivity of the map B can be obtained by using %Elie Cartan �Car angularinvariant A �x�� A �x� � �� for a triple x � �x�� x�� x�� of points in �H �

C�

It is known �see �Go�� that� for two triples x and y� A �x� � A �y� if and only if thereexists g � PU� � � such that y � g�x�" furthermore� such a g is unique providedthat A �x� is neither zero nor �� Here A �x� � � if and only if x�� x� and x�

lie on an R�circle� and A �x� � �� if and only if x�� x� and x� lie on a chain�C �circle��Namely� let g� � GnG� be a generator of the group G in �� whose �xed point

x� � ��G� lies in R� f�g � H� and x�� G�� the corresponding �xed point


of the element ��g�� G� under the free�product isomorphism �� G � G��Due to our construction� one can see that the orbit ��x�� G�� under theloxodromic �dilation� subgroup G� � G � G� approximates the origin along a ray�� which has a non�zero angle with the ray R� f�g � H� The latter rayalso contains an orbit ��x�� G�� of a limit point x

� of G� which approximatesthe origin from the other side� Taking triples x � �x�� x�� and x� � �x�� x��of points which lie correspondingly in the limit sets ��G� and ��G�� we have thatA �x� � � and A �x� � �� Due to Theorem �� both limit sets are topologicalcircles which however cannot be equivalent under a hyperbolic isometry because ofdi�erent Cartan invariants �and hence� again� our deformation is not trivial��

Similarly� for two di�erent values and �� we have triples x� and x�� withdi�erent �non�trivial� Cartan angular invariants A �x� � �� A �x�� Hence ��G�� and��G�� are not PU� � ��equivalent�

�

One can apply the above proof to a general situation of bending deformations ofa complex hyperbolic surface M � H

�C�G whose holonomy group G � PU� � � has

a non�elementary subgroup Gr preserving a totally real geodesic plane H�R� In other

words� such a complex surfaces M has an embedded totally real geodesic surfacewith geodesic boundary� In particular all complex surfaces constructed in �GKLwith a given Toledo invariant lie in this class� So we immediately have�

Corollary �� Let M � H �C�G be a complex hyperbolic surface with embedded

totally real geodesic surface Sr � M with geodesic boundary� and B � �� D�M� be the bending deformation of M along a simple closed geodesic � � Sr�Then the map � � B � �� T �M� � D�M��PU� � � is a smooth embeddingprovided that the limit set ��G� of the holonomy group G does not belong to theG�orbit of the real circle S�

Rand the chain S�

C� where the latter is the in�nity of the

complex geodesic containing a lift '� � H�Cof the closed geodesic �� and the former

one contains the limit set of the holonomy group Gr � G of the geodesic surfaceSr�

�

As an application of the constructed bending deformations� we answer a wellknown question about cusp groups on the boundary of the Teichm�uller space T �M�of a Stein complex hyperbolic surface M �bering over a compact Riemann surfaceof genus p � � It is a direct corollary of the following result� see �AG�

Theorem �� Let G � PO� � � � PU� � � be a non�elementary discrete groupSp of genus p � � Then� for any simple closed geodesic � in the Riemann surfaceS � H�

R�G� there is a continuous deformation �t � f�t induced by G�equivariant

quasiconformal homeomorphisms ft � H �C� H �

Cwhose limit representation ��

corresponds to a boundary cusp point of the Teichm�uller space T �G�� that is� theboundary group ��G� has an accidental parabolic element ��g�� where g� � Grepresents the geodesic � � S�


We note that� due to our construction of such continuous quasiconformal defor�mations in �AG� they are independent if the corresponding geodesics �i � Sp aredisjoint� It implies the existence of a boundary group in �T �G� with �maximal�number of non�conjugate accidental parabolic subgroups�

Corollary �� Let G � PO� � � � PU� � � be a uniform lattice isomorphicto the fundamental group of a closed surface Sp of genus p � � Then there is acontinuous deformation R � R�p�� T �G� whose boundary group G� � R��G�has ��p� �� non�conjugate accidental parabolic subgroups�

Finally� we mention another aspect of the intrigue Problem �� on geometrical�niteness of complex hyperbolic surfaces �see �AX� AX � for which it may perhapsbe possible to apply our complex bending deformations�

Problem� Construct a geometrically in�nite ��nitely generated� discrete groupG � PU� � � whose limit set is the whole sphere at in�nity� ��G� � �H �

C� H�

and which is the limit of convex cocompact groups Gi � PU� � � from the Te�ichm�uller space T �� of a convex cocompact group � � PU� � �� Is that possiblefor a Schottky group ��

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Geometry.and.topology.of.complex.hyperbolic.and.CR-manifolds