GEOMETRY AND TOPOLOGY OF COMPLEX

HYPERBOLIC AND CR-MANIFOLDS

Boris Apanasov

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 Kähler 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.

1. Introduction

This paper presents recent progress in studying topology and geometry of complex hyperbolic manifolds M with variable negative curvature and spherical Cauchy-Riemannian manifolds with Carnot-Caratheodory structure at infinity M_{∞}.

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 Kähler 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 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 [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 [Th, A1, A2] and §7) as well as the rigidity of quaternionic/octionic hyperbolic manifolds and higher-rank locally symmetric spaces [MG1, Co2, P]. Finally, since

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 57, 55, 53, 51, 32, 22, 20.

Key words and phrases. Complex hyperbolic geometry, Cauchy-Riemannian manifolds, discrete groups, geometrical finiteness, nilpotent and Heisenberg groups, Bieberbach theorems, fiber bundles, homology cobordisms, quasiconformal maps, structure deformations, Teichmüller spaces.

Research in MSRI was supported in part by NSF grant DMS-9022140.

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 [W, LB, BE, FS, S, A9] and §5.

Complex hyperbolic geometry is the geometry of the unit ball 𝔹_{ℂ}^{n} in ℂ^{n} with the Kähler structure given by the Bergman metric (compare [CG, Go3], whose automorphisms are biholomorphic automorphisms of the ball, i.e., elements of P U(n, 1). (We notice that complex hyperbolic manifolds with non-elementary fundamental groups are complex hyperbolic in the sense of S.Kobayashi [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 ℍ_{ℂ}^{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 H_{n}), which generalizes a Bieberbach theorem 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 finiteness condition on its fundamental group π_{1}(M) = G ⊂ P U(n, 1), which in particular implies that G is finitely generated [Bow] and even finitely presented, see Corollary 4.5. The original definition of a geometrically finite manifold M (due to L.Ahlfors [ 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 [BM, MA, Th, A1, A3]. Some of those ideas also work in spaces with pinched negative curvature, see [Bow]. However, geometric methods based on consideration of finite sided fundamental polyhedra cannot be used in spaces of variable curvature, see §4, 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 [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 Kähler 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 [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 [EMM] and to D.Burns theorem in the case when only one boundary component at infinity is compact (see also [NR1, Th.4.4], [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_{ℂ} M ≥ 3) if one of the above compactness conditions holds. However, if boundary components of M are non-compact, the boundary ∂_{∞}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 π_{1}(M) into P U(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 P U(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 [Mo1] (for all locally symmetric spaces of rank one). Nevertheless their constant curvature analogue, real hyperbolic manifolds are flexible in low dimensions and in the sense of quasi-Fuchsian deformations (see our discussion in §7). Contrasting to such a flexibility, complex hyperbolic manifolds share the super-rigidity of quaternionic/octionic hyperbolic manifolds (see Pansu’s [P] and Corlette’s [Co1-2] rigidity theorems, analogous to Margulis’s [MG1] super-rigidity in higher rank). Namely, due to Goldman’s [Go1] local rigidity theorem in dimension n = 2 and its extension [GM] for n ≥ 3, every nearby discrete representation ρ : G → P U(n, 1) of a cocompact lattice G ⊂ P U(n − 1, 1) stabilizes a complex totally geodesic subspace ℍ_{ℂ}^{n−1} in ℍ_{ℂ}^{n}, and for n ≥ 3, this rigidity is even global due to a celebrated Yue’s theorem [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 ⊂ P U(n, 1) preserves a complex totally geodesic hyperspace in ℍ_{ℂ}^{n} is essential for local rigidity of deformations only for co-compact lattices G ⊂ P U(n − 1, 1). This is due to the following our result [ACG]:

Theorem 7.1. Let G ⊂ P U(1, 1) be a co-finite free lattice whose action in ℍ_{ℂ}^{2} is generated by four real involutions (with fixed mutually tangent real circles at infinity). Then there is a continuous family {f_{α}}, −ϵ < α < ϵ, of G-equivariant homeomorphisms in \overline{ℍ_{ℂ}^{2}} which induce non-trivial quasi-Fuchsian (discrete faithful) representations f_{α}^{∗} : G → P U(2, 1). Moreover, for each α ≠ 0, any G-equivariant homeomorphism of \overline{ℍ_{ℂ}^{2}} that induces the representation f_{α}^{∗} cannot be quasiconformal.

This also shows the impossibility to extend the Sullivan’s quasiconformal stability theorem [Su2] to that situation, as well as provides the first continuous (topological) deformation of a co-finite Fuchsian group G ⊂ P U(1, 1) into quasi-Fuchsian groups G_{α} = f_{α}Gf_{α}^{−1} ⊂ P U(2, 1) with the (arbitrarily close to one) Hausdorff dimension dim_{H} Λ(G_{α}) > 1 of the limit set Λ(G_{α}), α ≠ 1, compare [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 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:

Theorem 7.2. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a given (non-elementary) discrete group. Then, for any simple closed geodesic α in the Riemann 2-surface S = H_{ℝ}^{2}/G and a sufficiently small η_{0} > 0, there is a holomorphic family of G-equivariant quasiconformal homeomorphisms F_{η} : \overline{ℍ_{ℂ}^{2}} → \overline{ℍ_{ℂ}^{2}}, −η_{0} < η < η_{0}, which defines the bending (quasi-Fuchsian) deformation B_{α} : (−η_{0}, η_{0}) → R_{0}(G) of the group G along the geodesic α, B_{α}(η) = F_{η}^{∗}.

The constructed deformations depend on many parameters described by the Teichmüller space T (M) of isotopy classes of complex hyperbolic structures on M, or equivalently by the Teichmüller space T (G) = R_{0}(G)/P U(n, 1) of conjugacy classes of discrete faithful representations ρ ∈ R_{0}(G) ⊂ Hom(G, P U(n, 1)) of G = π_{1}(M):

Corollary 7.3. Let S_{p} = ℍ_{ℝ}^{2}/G be a closed totally real geodesic surface of genus p > 1 in a given complex hyperbolic surface M = ℍ_{ℂ}^{2}/G, G ⊂ P O(2, 1) ⊂ P U(2, 1). Then there is an embedding π ∘ B : B^{3p−3} ↪ T (M) of a real (3p − 3)-ball into the Teichmüller space of M, defined by bending deformations along disjoint closed geodesics in M and by the projection π : D(M) → T (M) = D(M)/P U(2, 1) in the development space D(M).

As an application of the constructed deformations, we answer a well known question about cusp groups on the boundary of the Teichmüller space T (M) of a (Stein) complex hyperbolic surface M fibering over a compact Riemann surface of genus p > 1 [AG]:

Corollary 7.12. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a uniform lattice isomorphic to the fundamental group of a closed surface S_{p} of genus p ≥ 2. Then there is a continuous deformation R : ℝ^{3p−3} → T (G) (induced by G-equivariant quasiconformal homeomorphisms of ℍ_{ℂ}^{2}) whose boundary group G_{∞} = R(∞)(G) has (3p − 3) non-conjugate accidental parabolic subgroups.

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 [A7]:

Theorem 6.2. Let ϕ : G → H be a type preserving isomorphism of two non-ele-mentary geometrically finite groups G, H ⊂ P U(n, 1). Then there exists a unique equivariant homeomorphism f_{ϕ} : Λ(G) → Λ(H) of their limit sets that induces the isomorphism ϕ. Moreover, if Λ(G) = ∂ℍ_{ℂ}^{n}, the homeomorphism f_{ϕ} is the restriction of a hyperbolic isometry h ∈ P U(n, 1).

We note that, in contrast to Tukia [Tu] isomorphism theorem in the real hyperbolic geometry, one might suspect that in general the homeomorphism f_{ϕ} has no good metric properties, compare Theorem 7.1. This is still one of open problems in complex hyperbolic geometry (see §6 for discussions).

2. Complex hyperbolic and Heisenberg manifolds

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 Kähler geometry of the complex hyperbolic space (compare [GP1, Go3, KR]).

One can realize the complex hyperbolic geometry in the complex projective space,

ℍ_{ℂ}^{n} = {[z] ∈ ℂℙ^{n} : ⟨z, z⟩ < 0 , z ∈ ℂ^{n,1}} ,

as the set of negative lines in the Hermitian vector space ℂ^{n,1}, with Hermitian structure given by the indefinite (n, 1)-form ⟨z, w⟩ = z_{1}\overline{w}_{1} +⋯+z_{n}\overline{w}_{n} −z_{n+1}\overline{w}_{n+1}. Its boundary ∂ℍ_{ℂ}^{n} = {[z] ∈ ℂℙ^{n,1} : ⟨z, z⟩ = 0} consists of all null lines in ℂℙ^{n} and is homeomorphic to the (2n-1)-sphere S^{2n−1}.

The full group Isom ℍ_{ℂ}^{n} of isometries of ℍ_{ℂ}^{n} is generated by the group of holomorphic automorphisms (= the projective unitary group P U(n, 1) defined by the group U(n, 1) of unitary automorphisms of ℂ^{n,1}) together with the antiholomorphic automorphism of ℍ_{ℂ}^{n} defined by the ℂ-antilinear unitary automorphism of ℂ^{n,1} given by complex conjugation z ↦ z̄. The group P U(n, 1) can be embedded in a linear group due to A.Borel [Bor] (cf. [AX1, L.2.1]), hence any finitely generated group G ⊂ P U(n, 1) is residually finite and has a finite index torsion free subgroup. Elements g ∈ P U(n, 1) are of the following three types. If g fixes a point in ℍ_{ℂ}^{n}, it is called elliptic. If g has exactly one fixed point, and it lies in ∂ℍ_{ℂ}^{n}, g is called parabolic. If g has exactly two fixed points, and they lie in ∂ℍ_{ℂ}^{n}, g is called loxodromic. These three types exhaust all the possibilities.

There are two common models of complex hyperbolic space ℍ_{ℂ}^{n} as domains in ℂ^{n}, the unit ball 𝔹_{ℂ}^{n} and the Siegel domain 𝔖_{n}. They arise from two affine patches in projective space related to ℍ_{ℂ}^{n} and its boundary. Namely, embedding ℂ^{n} onto the affine patch of ℂℙ^{n,1} defined by z_{n+1} ≠ 0 (in homogeneous coordinates) as A : ℂ^{n} → ℂℙ^{n}, z ↦ [(z, 1)], we may identify the unit ball 𝔹_{ℂ}^{n}(0, 1) ⊂ ℂ^{n} with ℍ_{ℂ}^{n} = A(𝔹_{ℂ}^{n}). Here the metric in ℂ^{n} is defined by the standard Hermitian form ⟨⟨ , ⟩⟩, and the induced metric on 𝔹_{ℂ}^{n} is the Bergman metric (with constant holomorphic curvature -1) whose sectional curvature is between -1 and -1/4.

The Siegel domain model of ℍ_{ℂ}^{n} arises from the affine patch complimentary to a projective hyperplane H_{∞} which is tangent to ∂ℍ_{ℂ}^{n} at a point ∞ ∈ ∂ℍ_{ℂ}^{n}. For example, taking that point ∞ as (0′, −1, 1) with 0′ ∈ ℂ^{n−1} and H_{∞} = {[z] ∈ ℂℙ^{n} : z_{n} + z_{n+1} = 0}, one has the map S : ℂ^{n} → ℂℙ^{n}\H_{∞} such that

( \begin{matrix}z′ \\ z_{n} \end{matrix} ) ⟼ [ \begin{matrix}z′ \\ \frac{1}{2} − z_{n} \\ \frac{1}{2} + z_{n} \end{matrix} ] where z′ = ( z_{1} ; ⋮ ; z_{n−1} ) ∈ ℂ^{n−1} .

In the obtained affine coordinates, the complex hyperbolic space is identified with the Siegel domain

𝔖_{n} = S^{−1}(ℍ_{ℂ}^{n}) = \{z ∈ ℂ^{n} : z_{n} + \overline{z}_{n} > ⟨⟨z′, z′⟩⟩\} ,

where the Hermitian form is ⟨S(z), S(w)⟩ = ⟨⟨z′, w′⟩⟩ − z_{n} − \overline{w}_{n}. The automorphism group of this affine model of ℍ_{ℂ}^{n} is the group of affine transformations of ℂ^{n} preserving 𝔖_{n}. Its unipotent radical is the Heisenberg group H_{n} consisting of all Heisenberg translations

T_{ξ,v} : (w′, w_{n}) ↦ (w′ + ξ, w_{n} + ⟨⟨ξ, w′⟩⟩ + \frac{1}{2}(⟨⟨ξ, xi⟩⟩ − iv)) ,

where w′, ξ ∈ ℂ^{n−1} and v ∈ ℝ.

In particular H_{n} acts simply transitively on ∂ℍ_{ℂ}^{n}\{∞}, and one obtains the upper half space model for complex hyperbolic space ℍ_{ℂ}^{n} by identifying ℂ^{n−1} × ℝ × [0, ∞) and \overline{ℍ_{ℂ}^{n}}\\{∞\} as

(ξ, v, u) ⟼ [ \begin{matrix}ξ \\ \frac{1}{2}(1 − ⟨⟨ξ, ξ⟩⟩ − u + iv) \\ \frac{1}{2}(1 + ⟨⟨ξ, ξ⟩⟩ + u − iv) \end{matrix} ] ,

where (ξ, v, u) ∈ ℂ^{n−1} × ℝ × [0, ∞) are the horospherical coordinates of the corresponding point in \overline{ℍ_{ℂ}^{n}}\\{∞\} (with respect to the point ∞ ∈ ∂ℍ_{ℂ}^{n}, see [GP1]).

We notice that, under this identification, the horospheres in ℍ_{ℂ}^{n} centered at ∞ are the horizontal slices H_{t} = {(ξ, v, u) ∈ ℂ^{n−1} × ℝ × ℝ_{+} : u = t}, and the geodesics running to ∞ are the vertical lines c_{ξ,v}(t) = (ξ, v, e^{2t}) passing through points (ξ, v) ∈ ℂ^{n−1} × ℝ. Thus we see that, via the geodesic perspective from ∞, various horospheres correspond as H_{t} → H_{u} with (ξ, v, t) ↦ (ξ, v, u).

The “boundary plane” ℂ^{n−1} × ℝ × {0} = ∂ℍ_{ℂ}^{n}\{∞} and the horospheres H_{u} = ℂ^{n−1} ×ℝ×{u}, 0 < u < ∞, centered at ∞ are identified with the Heisenberg group H_{n} = ℂ^{n−1} × ℝ. It is a 2-step nilpotent group with center {0} × ℝ ⊂ ℂ^{n−1} × ℝ, with the isometric action on itself and on ℍ_{ℂ}^{n} by left translations:

T_{(ξ_{0},v_{0})} : (ξ, v, u) ⟼ (ξ_{0} + ξ , v_{0} + v + 2 Im⟨⟨ξ_{0}, ξ⟩⟩ , u) ,

and the inverse of (ξ, v) is (ξ, v)^{−1} = (−ξ, −v). The unitary group U(n − 1) acts on H_{n} and ℍ_{ℂ}^{n} by rotations: A(ξ, v, u) = (Aξ , v , u) for A ∈ U(n − 1). The semidirect product H(n) = H_{n} ⋊ U(n − 1) is naturally embedded in U(n, 1) as follows:

A ⟼ ( \begin{matrix}A & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ∈ U(n, 1) for A ∈ U(n − 1) ,

(ξ, v) ⟼ ( \begin{matrix}I_{n−1} & ξ & ξ \\ −ξ̄^{t} & 1 − \frac{1}{2}(|ξ|^{2} − iv) & −\frac{1}{2}(|ξ|^{2} − iv) \\ ξ̄^{t} & \frac{1}{2}(|ξ|^{2} − iv) & 1 + \frac{1}{2}(|ξ|^{2} − iv) \end{matrix} ) ∈ U(n, 1)

where (ξ, v) ∈ H_{n} = ℂ^{n−1} × ℝ and ξ̄^{t} is the conjugate transpose of ξ.

The action of H(n) on \overline{ℍ_{ℂ}^{n}}\\{∞\} also preserves the Cygan metric ρ_{c} there, which plays the same role as the Euclidean metric does on the upper half-space model of the real hyperbolic space ℍ^{n} and is induced by the following norm:

||(ξ, v, u)||_{c} = | ||ξ||^{2} + u − iv|^{1/2} , (ξ, v, u) ∈ ℂ^{n−1} × ℝ × [0, ∞) .  (2.1)

The relevant geometry on each horosphere H_{u} ⊂ ℍ_{ℂ}^{n}, H_{u} ≅ H_{n} = ℂ^{n−1} × ℝ, is the spherical CR-geometry induced by the complex hyperbolic structure. The geodesic perspective from ∞ defines CR-maps between horospheres, which extend to CR-maps between the one-point compactifications H_{u} ∪ ∞ ≈ S^{2n−1}. In the limit, the induced metrics on horospheres fail to converge but the CR-structure remains fixed. In this way, the complex hyperbolic geometry induces CR-geometry on the sphere at infinity ∂ℍ_{ℂ}^{n} ≈ S^{2n−1}, naturally identified with the one-point compactification of the Heisenberg group H_{n}.

3. Discrete actions on nilpotent groups and Heisenberg manifolds

In order to study the structure of Heisenberg manifolds (i.e., the manifolds locally modeled on the Heisenberg group H_{n}) and cusp ends of complex hyperbolic manifolds, we need a Bieberbach type structural theorem for isometric discrete group actions on H_{n}, originally proved in [AX1]. It claims that each discrete isometry group of the Heisenberg group H_{n} preserves some left coset of a connected Lie subgroup, on which the group action is cocompact.

Here we consider more general situation. Let N be a connected, simply connected nilpotent Lie group, C a compact group of automorphisms of N, and Γ a discrete subgroup of the semidirect product N ⋊ C. Such discrete groups are the holonomy groups of parabolic ends of locally symmetric rank one (negatively curved) manifolds and can be described as follows.

Theorem 3.1. There exist a connected Lie subgroup V of N and a finite index normal subgroup Γ^{∗} of Γ with the following properties:

(1) There exists b ∈ N such that bΓb^{−1} preserves V.

(2) V/bΓb^{−1} is compact.

(3) bΓ^{∗}b^{−1} acts on V by left translations and this action is free.

Remark 3.2. (1) It immediately follows that any discrete subgroup Γ ⊂ N ⋊ C is virtually nilpotent because it has a finite index subgroup Γ^{∗} ⊂ Γ isomorphic to a lattice in V ⊂ N.

(2) Here, compactness of C is an essential condition because of Margulis [MG2] construction of nonabelian free discrete subgroups Γ of R^{3} ⋊ GL(3, R).

(3) This theorem generalizes a Bieberbach theorem for Euclidean spaces, see [Wo], and strengthens a result by L.Auslander [Au] who claimed those properties not for whole group Γ but only for its finite index subgroup. Initially in [AX1], we proved this theorem for the Heisenberg group H_{n} where we used Margulis Lemma [MG1, BGS] and geometry of H_{n} in order to extend the classical arguments in [Wo]. In the case of general nilpotent groups, our proof uses different ideas and goes as follows (see[AX2] for details).

Sketch of Proof. Let p : Γ → C be the composition of the inclusion Γ ⊂ N ⋊ C and the projection N ⋊ C → C, G the identity component of \overline{ΓN}, and Γ_{1} = G ∩ Γ. Due to compactness of C, G has finite index in \overline{ΓN}, so Γ_{1} has finite index in Γ. Let W ⊂ N be the analytic subgroup pointwise fixed by p(Γ_{1}). Due to [Au], for all γ = (w, c) ∈ Γ_{1}, w lies in W. Thus Γ preserves W and, by replacing N with W, we may assume that Φ = p(Γ) is finite.

Consider Γ^{∗} = ker(p) which is a discrete subgroup of N and has finite index in Γ. Let V be the connected Lie subgroup of N in which Γ^{∗} is a lattice. Then the conjugation action of Γ on Γ^{∗} induces a Γ-action on V. We form the semi-direct product V ⋊ Γ and let K = {(a^{−1}, (a, 1)) ∈ V ⋊ Γ : (a, 1) ∈ Γ^{∗}}. Obviously, K is a normal subgroup of V ⋊ Γ. Defining the maps i : V → V ⋊ Γ/K by i(v) = (v, (1, 1))K and π : V ⋊ Γ/K → Φ by π(v, (a, A)) = A, we get a short exact sequence

1 ⟶ V ⟶^{i} V ⋊ Γ/K ⟶^{π} Φ ⟶ 1 .

Since any extension of a finite group by a simply connected nilpotent Lie group splits, there is a homomorphism s : Φ → V ⋊ Γ/K such that π ∘ s = id_{Φ}. For each A ∈ Φ, we fix an element (f(A), (g(A), A)) ∈ V ⋊ Γ representing s(A). Since s is a homomorphism, we have

g(AB)^{−1}f(AB)^{−1} = A (g(B)^{−1}f(B)^{−1}) g(A)^{−1}f(A)^{−1} for A, B ∈ Φ . (4.3)

Define h : Φ → N by h(A) = g(A)^{−1}f(A)^{−1}. Then (2.4) shows that h is a cocycle. Since Φ is finite and N is a simply connected nilpotent Lie group, H^{1}(Φ, N) = 0 due to [LR]. Thus there exists b ∈ N such that h(A) = A(b^{−1})b for all A ∈ Φ.

On the other hand, π((1, (a, A))K) = π((f(A), (g(A), A))K) = A for any γ = (a, A) ∈ Γ. It follows that there is v_{0} ∈ 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_{0}b^{−1}.

Now consider the group bΓb^{−1} which acts on bV b^{−1}., For any γ = (a, A) ∈ Γ, the action of the element bγb^{−1} = (baA(b^{−1}), A) on bV b^{−1} is as follows:

((baA(b^{−1}), A), v′) → baA(b^{−1})A(v′)(baA(b^{−1}))^{−1} .

In particular, baA(b^{−1})·A(bV b^{−1})·(baA(b^{−1}))^{−1} = bV b^{−1}. Therefore, A(bV b^{−1}) = bV b^{−1} because of baA(b^{−1}) = bv_{0}b^{−1} ∈ bV b^{−1}, and hence bγb^{−1} preserves bV b^{−1}.

□

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 (H_{n}, H(n))-geometry and each of them can be represented as the quotient H_{n}/G under a discrete, free isometric action of its fundamental group G on H_{n}, i.e., the isometric action of a torsion free discrete subgroup of H(n) = H_{n} ⋊ U(n − 1). Actually, we establish fiber bundle structures on all noncompact Heisenberg manifolds:

Theorem 3.3. Let Γ ⊂ H_{n} ⋊ U(n − 1) be a torsion-free discrete group acting on the Heisenberg group H_{n} = ℂ^{n−1} ×ℝ with non-compact quotient. Then the quotient H_{n}/Γ 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.

The proof of this claim (see [AX1]) is based on two facts due to Theorem 3.1. First, that the discrete holonomy group Γ ≅ π_{1}(M) of any noncompact Heisenberg manifold M = H_{n}/Γ, Γ ⊂ H(n), has a proper Γ-invariant subspace H_{Γ} ⊂ H_{n}. And second, the compact manifold H_{Γ}/Γ is finitely covered by H_{Γ}/Γ^{∗} where Γ^{∗} acts on H_{Γ} by translations. The structure of the covering manifold H_{G}/G^{∗} is given in the following lemma.

Lemma 3.4. Let V be a connected Lie subgroup of the Heisenberg group H_{n} and G ⊂ V a discrete co-compact subgroup of V. Then the manifold V/G is

(1) a torus if V is Abelian;

(2) the total space of a torus bundle over a torus if V is not Abelian.

Though noncompact Heisenberg manifolds M are vector bundles H_{n}/Γ → H_{Γ}/Γ, simple examples show [AX1] that such vector bundles may be non-trivial in general. However, up to finite coverings, they are trivial [AX1]:

Theorem 3.5. Let Γ ⊂ H_{n}⋊U(n−1) be a discrete group and H_{Γ} ⊂ H_{n} a connected Γ-invariant Lie subgroup on which Γ acts co-compactly. Then there exists a finite index subgroup Γ_{0} ⊂ Γ such that the vector bundle H_{n}/Γ_{0} → H_{Γ}/Γ_{0} is trivial. In particular, any Heisenberg orbifold H_{n}/Γ is finitely covered by the product of a compact nil-manifold H_{Γ}/Γ_{0} and an Euclidean space.

We remark that in the case when Γ ⊂ H_{n} ⋊ U(n − 1) is a lattice, that is the quotient H_{n}/Γ is compact, the existence of such finite cover of H_{n}/Γ by a closed nilpotent manifold H_{n}/Γ_{0} is due to Gromov [Gr] and Buser-Karcher [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 Γ is torsion-free. After that, we shall find a finite index subgroup Γ_{0} ⊂ Γ whose rotational part is “good”. Then we shall express the vector bundle H_{n}/Γ_{0} → H_{Γ}/Γ_{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:

Lemma 3.6. Let F ×_{H} V be a fiber product and suppose that the homomorphism ρ : H → GL(V) extends to a homomorphism ρ : F → GL(V). Then F ×_{H} V is a trivial bundle, F ×_{H} V ≅ F/H × V.

Proof. The isomorphism F ×_{H}V ≅ F/H×V is given by [f, v] → (Hf, ρ(f)^{−1}(v)). □

4. Geometrical finiteness in complex hyperbolic geometry

Our main assumption on a complex hyperbolic n-manifold M is the geometrical finiteness of its fundamental group π_{1}(M) = G ⊂ P U(n, 1), which in particular implies that the discrete group G is finitely generated.

Here a subgroup G ⊂ P U(n, 1) is called discrete if it is a discrete subset of P U(n, 1). The limit set Λ(G) ⊂ ∂ℍ_{ℂ}^{n} of a discrete group G is the set of accumulation points of (any) orbit G(y), y ∈ ℍ_{ℂ}^{n}. The complement of Λ(G) in ∂ℍ_{ℂ}^{n} is called the discontinuity set Ω(G). A discrete group G is called elementary if its limit set Λ(G) consists of at most two points. An infinite discrete group G is called parabolic if it has exactly one fixed point fix(G); then Λ(G) = fix(G), and G consists of either parabolic 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 with as it is in the constant curvature space. However the results of §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 = ℍ^{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 [Ah]. However, we can define geometrically finite groups G ⊂ P U(n, 1) as those ones whose limit sets Λ(G) consist of only conical limit points and parabolic (cusp) points p with compact quotients (Λ(G)\{p})/G_{p} with respect to parabolic stabilizers G_{p} ⊂ G of p, see [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 ℍ^{n} (see [Mar, Th, A1, A3]) but also in spaces with variable pinched negative curvature [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 ⊂ P U(n, 1).

Namely, suppose a point p ∈ ∂ℍ_{ℂ}^{n} is fixed by some parabolic element of a given discrete group G ⊂ P U(n, 1), and G_{p} is the stabilizer of p in G. Conjugating G by an element h_{p} ∈ P U(n, 1) , h_{p}(p) = ∞, we may assume that the stabilizer G_{p} is a subgroup G_{∞} ⊂ H(n). In particular, if p is the origin 0 ∈ H_{n}, the transformation h_{0} can be taken as the Heisenberg inversion I in the hyperchain ∂ℍ_{ℂ}^{n−1}. It preserves the unit Heisenberg sphere S_{c}(0, 1) = {(ξ, v) ∈ H_{n} : ||(ξ, v)||_{c} = 1} and acts in H_{n} as follows:

I(ξ, v) = (\frac{ξ}{|ξ|^{2} − iv} , \frac{−v}{v^{2} + |ξ|^{4}} ) where (ξ, v) ∈ H_{n} = ℂ^{n−1} × ℝ .  (4.1)

For any other point p, we may take h_{p} as the Heisenberg inversion I_{p} which preserves the unit Heisenberg sphere S_{c}(p, 1) = {(ξ, v) : ρ_{c}(p, (ξ, v)) = 1} centered at p. The inversion I_{p} is conjugate of I by the Heisenberg translation T_{p} and maps p to ∞.

After such a conjugation, we can apply Theorem 3.1 to the parabolic stabilizer G_{∞} ⊂ H(n) and get a connected Lie subgroup H_{∞} ⊆ H_{n} preserved by G_{∞} (up to changing the origin). So we can make the following definition.

Definition 4.2. A set U_{p,r} ⊂ \overline{ℍ_{ℂ}^{n}}\\{p\} is called a standard cusp neighborhood of radius r > 0 at a parabolic fixed point p ∈ ∂ℍ_{ℂ}^{n} of a discrete group G ⊂ P U(n, 1) if, for the Heisenberg inversion I_{p} ∈ P U(n, 1) with respect to the unit sphere S_{c}(p, 1), I_{p}(p) = ∞ , the following conditions hold:

(1) U_{p,r} = I_{p}^{−1} ({x ∈ ℍ_{ℂ}^{n} ∪ H_{n} : ρ_{c}(x, H_{∞}) ≥ 1/r}) ;

(2) U_{p,r} is precisely invariant with respect to G_{p} ⊂ G, that is:

γ(U_{p,r}) = U_{p,r} for γ ∈ G_{p} and g(U_{p,r}) ∩ U_{p,r} = ∅ for g ∈ G\G_{p} .

A parabolic point p ∈ ∂ℍ_{ℂ}^{n} of G ⊂ P U(n, 1) is called a cusp point if it has a cusp neighborhood U_{p,r}.

We remark that some parabolic points of a discrete group G ⊂ P U(n, 1) may not be cusp points, see examples in §5.4 of [AX1]. Applying Theorem 3.1 and [Bow], we have:

Lemma 4.3. Let p ∈ ∂ℍ_{ℂ}^{n} be a parabolic fixed point of a discrete subgroup G in P U(n, 1). Then p is a cusp point if and only if (Λ(G)\{p})/G_{p} is compact.

This and finiteness results of Bowditch [B] allow us to use another equivalent definitions of geometrical finiteness. In particular it follows that a discrete subgroup G in P U(n, 1) is geometrically finite if and only if its quotient space M(G) = [ℍ_{ℂ}^{n} ∪ Ω(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} ≈ (S_{p,r}/G_{p}) × (0, 1] ,  (4.4)

where

S_{p,r} = ∂_{H}U_{p,r} = I_{p}^{−1} ({x ∈ H_{ℂ}^{n} ∪ H_{n} : ρ_{c}(x, H_{∞}) = 1/r}) .

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 ≤ 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:

Corollary 4.5. Geometrically finite groups G ⊂ P U(n, 1) are finitely presented.

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 ⊂ P U(n, 1) bounded by bisectors in a complicated way, see [ Mo2, GP1, FG]. In the case of discrete parabolic groups G ⊂ P U(n, 1), one may expect that the Dirichlet polyhedron D_{y}(G) centered at a point y lying in a Ginvariant subspace has finitely many sides. It is true for real hyperbolic spaces [A1] as well as for cyclic and dihedral parabolic groups in complex hyperbolic spaces. Namely, due to [ Ph], Dirichlet polyhedra D_{y}(G) are always two sided for any cyclic group G ⊂ P U(n, 1) generated by a Heisenberg translation. Due the main result in [GP1], this finiteness also holds for a cyclic ellipto-parabolic group or a dihedral parabolic group G ⊂ P U(n, 1) generated by inversions in asymptotic complex hyperplanes in ℍ_{ℂ}^{n} if the central point y lies in a G-invariant vertical line or ℝ-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 [AX1]:

Theorem 4.6. For any discrete group G ⊂ P U(n, 1) generated by a parabolic element, there exists a point y_{0} ∈ ℍ_{ℂ}^{n} such that the Dirichlet polyhedron D_{y_{0}}(G) centered at y_{0} has two sides.

Proof. Conjugating G and due to Theorem 3.1, we may assume that G preserves a one dimensional subspace H_{∞} ⊂ H_{n} as well as H_{∞} × ℝ_{+} ⊂ ℍ_{ℂ}^{n}, where G acts by translations. So we can take any point y_{0} ∈ H_{∞} × ℝ_{+} 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.

□

However, the behavior of Dirichlet polyhedra for parabolic groups G ⊂ P U(n, 1) of rank more than one can be very bad. It is given by our construction [AX1], where we have evaluated intersections of Dirichlet bisectors with a 2-dimensional slice:

Theorem 4.7. Let G ⊂ P U(2, 1) be a discrete parabolic group conjugate to the subgroup Γ = {(m, n) ∈ ℂ × ℝ : m, n ∈ ℤ} of the Heisenberg group H_{2} = ℂ × ℝ. Then any Dirichlet polyhedron D_{y}(G) centered at any point y ∈ ℍ_{ℂ}^{2} has infinitely many sides.

Despite the above example, the below application of Theorem 3.1 provides a construction of fundamental polyhedra P (G) ⊂ ℍ_{ℂ}^{n} for arbitrary discrete parabolic groups G ⊂ P U(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 [A1] of finite sided pseudo-Dirichlet polyhedra in ℍ^{n} to the case of the complex hyperbolic space ℍ_{ℂ}^{n}.

Theorem 4.8. For any discrete parabolic group G ⊂ P U(n, 1), there exists a finite-sided fundamental polyhedron P (G) ⊂ ℍ_{ℂ}^{n}.

Proof. After conjugation, we may assume that G ⊂ H_{n} ⋊ U(n − 1). Let H_{∞} ⊂ H_{n} = ℂ^{n−1} × ℝ be the connected G-invariant subgroup given by Theorem 3.1. For a fixed u_{0} > 0, we consider the horocycle V_{u_{0}} = H_{∞} × {u_{0}} ⊂ ℂ^{n−1} × ℝ × ℝ_{+} = ℍ_{ℂ}^{n}. For distinct points y, y′ ∈ V_{u_{0}}, the bisector C(y, y′) = {z ∈ ℍ_{ℂ}^{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) = ⋂_{g∈G\{id}}{w ∈ ℍ_{ℂ}^{n} : d(w, y) < d(w, g(y))}

centered at a point y ∈ V_{u_{0}} is a fundamental polyhedron for the G-action on V_{u_{0}}. The polyhedron D_{y}(G)∩V_{u_{0}} is compact due to Theorem 3.3, and hence has finitely many sides. Now, considering G-equivariant projections [AX1]:

π : H_{n} → H_{∞} , π′ : ℍ_{ℂ}^{n} = H_{n} × ℝ_{+} → V_{u_{0}} , π′(x, u) = (π(x), u_{0}) ,

we get a finite-sided fundamental polyhedron π′^{−1}(D_{y}(G) ∩ V_{u_{0}}) for the action of G in ℍ_{ℂ}^{n}.

□

Another important application of Theorem 3.1 shows that cusp ends of a geometrically finite complex hyperbolic orbifolds M have, up to a finite covering of M, a very simple structure:

Theorem 4.9. Let G ⊂ P U(n, 1) be a geometrically finite discrete group. Then G has a subgroup G_{0} of finite index such that every parabolic subgroup of G_{0} is isomorphic to a discrete subgroup of the Heisenberg group H_{n} = ℂ^{n−1} × ℝ . In particular, each parabolic subgroup of G_{0} is free Abelian or 2-step nilpotent.

The proof of this fact [AX1] is based on the residual finiteness of geometrically finite subgroups in P U(n, 1) and the following two lemmas.

Lemma 4.10. Let G ⊂ H_{n} ⋊U(n−1) be a discrete group and H_{G} ⊂ H_{n} a minimal G-invariant connected Lie subgroup (given by Theorem 3.1). Then G acts on H_{G} by translations if G is either Abelian or 2-step nilpotent.

Lemma 4.11. Let G ⊂ H_{n} ⋊ U(n − 1) be a torsion free discrete group, F a finite group and ϕ : G ⟶ F an epimorphism. Then the rotational part of ker(ϕ) has strictly smaller order than that of G if one of the following happens:

(1) G contains a finite index Abelian subgroup and F is not Abelian;

(2) G contains a finite index 2-step nilpotent subgroup and F is not a 2-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 finiteness is very different in complex dimension two. Namely, it is a well known fact that any finitely generated discrete subgroup of P U(1, 1) or P O(2, 1) is geometrically finite. This and Goldman’s [ Go1] local rigidity theorem for cocompact lattices G ⊂ U(1, 1) ⊂ P U(2, 1) allow us to formulate the following conjecture:

Conjecture 4.12. All finitely generated discrete groups G ⊂ P U(2, 1) with nonempty discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{2} are geometrically finite.

5. 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_{ℂ}^{n} ∪ Ω(G)]/G with geometrically finite holonomy groups G ⊂ P U(n, 1). The boundary of this manifold, ∂M = Ω(G)/G, has a spherical CRstructure and, in general, is non-compact.

We are especially interested in the case of complex analytic surfaces, where powerful methods of 4-dimensional topology may be used. It is still unknown what are suitable cuts of 4-manifolds, which (conjecturally) split them into geometric blocks (alike Jaco-Shalen-Johannson decomposition of 3-manifolds in Thurston’s geometrization program; for a classification of 4-dimensional geometries, see [F, Wa]). Nevertheless, studying of complex surfaces suggests that in this case one can use integer homology 3-spheres and “almost flat” 3-manifolds (with virtually nilpotent fundamental groups). Actually, as Sections 3 and 4 show, the latter manifolds appear at the ends of finite volume complex hyperbolic manifolds. As it was shown by C.T.C.Wall [Wa], the assignment of the appropriate 4-geometry (when available) gives a detailed insight into the intrinsic structure of a complex surface. To identify complex hyperbolic blocks in such a splitting, one can use Yau’s uniformization theorem [Ya]. It implies that every smooth complex projective 2surface M with positive canonical bundle and satisfying the topological condition that χ(M) = 3 Signature(M), is a complex hyperbolic manifold. The necessity of homology sphere decomposition in dimension four is due to M.Freedman and L.Taylor result ([ FT]):

Let M be a simply connected 4-manifold with intersection form q_{M} which decomposes as a direct sum q_{M} = q_{M_{1}} ⊕ q_{M_{2}}, where M_{1}, M_{2} are smooth manifolds. Then the manifold M can be represented as a connected sum M = M_{1}#_{Σ}M_{2} along a homology sphere Σ.

Let us present an example of such a splitting, M = X#_{Σ}Y, of a simply connected complex surface M with the intersection form Q_{M} into smooth manifolds (with boundary) X and Y, along a ℤ-homology 3-sphere Σ such that Q_{M} = Q_{X} ⊕ Q_{Y}. Here one should mention that though X and Y are no longer closed manifolds, the intersection forms Q_{X} and Q_{Y} are well defined on the second cohomology and are unimodular due to the condition that Σ is a ℤ-homology 3-sphere.

Example 5.1. Let M be the Kummer surface

K3 = {[z_{0}, z_{1}, z_{2}, z_{3}] ∈ ℂℙ^{3} : z_{0}^{4} + z_{1}^{4} + z_{2}^{4} + z_{3}^{4} = 0} .

Then there are four disjointly embedded (Seifert fibered) ℤ-homology 3-spheres in M, which split the Kummer surface into five blocks:

K3 = X_{1} ∪_{Σ} Y_{1} ∪_{Σ′} Y_{2} ∪_{−Σ′} Y_{3} ∪_{−Σ} X_{2} ,

with intersection forms Q_{X_{j}} and Q_{Y_{i}} equal E_{8} and H, respectively:

E_{8} = ( \begin{matrix}−2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & −2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & −2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & −2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & −2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & −2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & −2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & −2 \end{matrix} ) , H = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Here the ℤ-homology spheres Σ and Σ′ are correspondingly the Poincaré homology sphere Σ(2, 3, 5) and Seifert fibered homology sphere Σ(2, 3, 7); the minus sign means the change of orientation.

Scheme of splitting. Due to J.Milnor [Mil] (see also [RV]), all Seifert fibered homology 3-spheres Σ can be seen as the boundaries at infinity of (geometrically finite) complex hyperbolic orbifolds ℍ_{ℂ}^{2}/Γ, where the fundamental groups π_{1}(Σ) = Γ ⊂ P U(2, 1) act free in the sphere at infinity ∂ℍ_{ℂ}^{2} = \overline{H_{2}}. In particular, the Seifert fibered homology sphere Σ′ = Σ(2, 3, 7) is diffeomorphic to the quotient [(ℂ × ℝ)\({0} × ℝ]/Γ(2, 3, 7). Here [(ℂ × ℝ)\({0} × ℝ)] is the complement in the 3-sphere \overline{H_{2}} = ∂B_{ℂ}^{2} to the boundary circle at infinity of the complex geodesic B_{ℂ}^{2} ∩ (ℂ × {0}), and the group Γ(2, 3, 7) ⊂ P U(2, 1) acts on this complex geodesic as the standard triangle group (2, 3, 7) in the disk Poincaré model of the hyperbolic 2-plane ℍ_{ℝ}^{2}.

This homology 3-sphere Σ′ embeds in the K3-surface M, splitting it into submanifolds with intersection forms E_{8}⊕H and E_{8}⊕2H. This embedding is described in [Lo] and [FS1]. One can keep decomposing the obtained two manifolds as in [FS2] and finally split it into five pieces. Among additional embedded homology spheres, there is the only one known homology 3-sphere with finite fundamental group, the Poincaré homology sphere Σ = Σ(2, 3, 5). One can introduce a spherical geometry on Σ by representing π_{1}(Σ) as a finite subgroup Γ(2, 3, 5) of the orthogonal group O(4) acting free on S^{3} = ∂B_{ℂ}^{2}. Then Σ(2, 3, 5) = S^{3}/Γ(2, 3, 5) can obtained by identifying the opposite sides of the spherical dodecahedron whose dihedral angles are 2π/3, see [KAG]. □

However we note that it is unknown whether the obtained blocks may support some homogeneous 4-geometries classified by Filipkiewicz [F] and (from the point of view of Kaḧler structures) C.T.C.Wall [Wa]. This raises a question whether homogeneous geometries or splitting along homology spheres (important from the topological point of view) are relevant for a geometrization of smooth 4-dimensional manifolds. For example, neither of Y_{i} blocks in Example 5.1 (with the intersection form H) can support a complex hyperbolic structure (which is a natural geometric candidate since Σ has a spherical CR-structure) because each of them has two compact boundary components.

In fact, in a sharp contrast to the real hyperbolic case, for a compact manifold M(G) (that is for a geometrically finite group G ⊂ P U(n, 1) without cusps), an application of Kohn-Rossi analytic extension theorem shows that the boundary of M(G) is connected, and the limit set Λ(G) is in some sense small (see [ EMM] and, for quaternionic and Caley hyperbolic manifolds [ C, CI]). Moreover, according to a recent result of D.Burns (see also Theorem 4.4 in [NR1]), the same claim about connectedness of the boundary ∂M(G) still holds if only a boundary component is compact. (In dimension n ≥ 3, D.Burns theorem based on [BuM] uses the last compactness condition to prove geometrical finiteness of the whole manifold M(G), see also [NR2].)

However, if no component of ∂M(G) is compact and we have no finiteness condition on the holonomy group of the complex hyperbolic manifold M(G), the situation is completely different due to our construction [AX1]:

Theorem 5.2. In any dimension n ≥ 2 and for any integers k, k_{0}, k ≥ k_{0} ≥ 0, there exists a complex hyperbolic n-manifold M = ℍ_{ℂ}^{n}/G, G ⊂ P U(n, 1), whose boundary at infinity splits up into k connected manifolds, ∂_{∞}M = N_{1} ∪ ⋯ ∪ N_{k}. Moreover, for each boundary component N_{j}, j ≤ k_{0}, its inclusion into the manifold M(G), i_{j} : N_{j} ⊂ M(G), induces a homotopy equivalence of N_{j} to M(G).

For a torsion free discrete group G ⊂ P U(n, 1), a connected component Ω_{0} of the discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{n} with the stabilizer G_{0} ⊂ G is contractible and G-invariant if and only if the inclusion N_{0} = Ω_{0}/G_{0} ⊂ M(G) induces a homotopy equivalence of N_{0} to M(G) [A1, AX1]. It allows us to reformulate Theorem 5.2 as

Theorem 5.3. In any complex dimension n ≥ 2 and for any natural numbers k and k_{0}, k ≥ k_{0} ≥ 0, there exists a discrete group G = G(n, k, k_{0}) ⊂ P U(n, 1) whose discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{n} splits up into k G-invariant components, Ω(G) = Ω_{1} ∪ ⋯ ∪ Ω_{k}, and the first k_{0} components are contractible.

Sketch of Proof. To prove this claim (see [AX1] for details), it is crucial to construct a discrete group G ⊂ P U(n, 1) whose discontinuity set consists of two G-invariant topological balls. To do that, we construct an infinite family Σ of disjoint closed Heisenberg balls B_{i} = B(a_{i}, r_{i}) ⊂ ∂ℍ_{ℂ}^{n} such that the complement of their closure, ∂ℍ_{ℂ}^{n} \ \overline{⋃_{i} B(a_{i}, r_{i})} = P_{1} ∪ P_{2}, consists of two topological balls, P_{1} and P_{2}. In our construction of such a family Σ of H-balls B_{j}, we essentially relie on the contact structure of the Heisenberg group H_{n}. Namely, Σ is the disjoint union of finite sets Σ_{i} of closed H-balls whose boundary H-spheres have “real hyperspheres” serving as the boundaries of (2n − 2)-dimensional cobordisms N_{i}. In the limit, these cobordisms converge to the set of limit vertices of the polyhedra P_{1} and P_{2} which are bounded by the H-spheres S_{j} = ∂B_{j}, B_{j} ∈ Σ. Then the desired group G = G(n, 2, 2) ⊂ P U(n, 1) is generated by involutions I_{j} which preserve those real (2n − 3)-spheres lying in S_{j} ⊂ ∂P_{1} ∪ ∂P_{2}, see Fig.1.

Figure 1. Cobordism N_{0} in H with two boundary real circles

We notice that, due to our construction, the intersection of each H-sphere S_{j} and each of the polyhedra P_{1} and P_{2} in the complement to the balls B_{j} ∈ Σ is a topological (2n − 2)-ball which splits into two sides, A_{j} and A′_{j}, and I_{i}(A_{i}) = A′_{i}. This allows us to define our desired discrete group G = G_{Σ} ⊂ P U(n, 1) as the discrete free product, G_{Σ} = ∗_{j} Γ_{j} = ∗_{i} ⟨I_{j}⟩, of infinitely many cyclic groups Γ_{j} generated by involutions I_{j} with respect to the H-spheres S_{j} = ∂B_{j}. So P_{1} ∪ P_{2} is a fundamental polyhedron for the action of G in ∂ℍ_{ℂ}^{n}, and sides of each of its connected components, P_{1} or P_{2}, are topological balls pairwise equivalent with respect to the corresponding generators I_{j} ∈ G. Applying standard arguments (see [A1], Lemmas 3.7, 3.8), we see that the discontinuity set Ω(G) ⊂ H_{n} consists of two G-invariant topological balls Ω_{1} and Ω_{2}, Ω_{k} = int (⋃_{g∈G} g(\overline{P_{k}})), k = 1, 2. The fact that Ω_{k} is a topological ball follows from the observation that this domain is the union of a monotone sequence,

V_{0} = int(\overline{P_{k}}) ⊂ V_{1} = int (\overline{P_{k}} ∪ I_{0}(\overline{P_{k}})) ⊂ V_{2} ⊂ …,

of open topological balls, see [Br]. Note that here we use the property of our construction that V_{i} is always a topological ball.

In the general case of k ≥ k_{0} ≥ 0, k ≥ 3, we can apply the above infinite free products and our cobordism construction of infinite families of H-balls with preassigned properties in order to (sufficiently closely) ”approximate” a given hypersurface in H_{n} by the limit sets of constructed discrete groups. For such hypersurfaces, we use the so called ”tree-like surfaces” which are boundaries of regular neighborhoods of trees in H_{n}. This allows us to generalize A.Tetenov’s [T1, KAG] construction of discrete groups G on the m-dimensional sphere S^{m} , m ≥ 3, whose discontinuity sets split into any given number k of G-invariant contractible connected components.

□

Although, in the general case of complex hyperbolic manifolds M with finitely generated π_{1}(M) ≅ G, the problem on the number of boundary components of M(G) is still unclear, we show below that the situation described in Theorem 5.3 is impossible if M is geometrically finite. We refer the reader to [AX1] for more precise formulation and proof of this cobordism theorem:

Theorem 5.4. Let G ⊂ P U(n, 1) be a geometrically finite non-elementary torsion free discrete group whose Kleinian manifold M(G) has non-compact boundary ∂M = Ω(G)/G with a component N_{0} ⊂ ∂M homotopy equivalent to M(G). Then there exists a compact homology cobordism M_{c} ⊂ M(G) such that M(G) can be reconstructed from M_{c} by gluing up a finite number of open collars M_{i} × [0, ∞) where each M_{i} is finitely covered by the product E_{k} ×B^{2n−k−1} of a closed (2n-1-k)-ball and a closed k-manifold E_{k} which is either flat or a nil-manifold (with 2-step nilpotent fundamental group).

In connection to this cobordism theorem, it is worth to mention another interesting fact due to Livingston–Myers [My] construction. Namely, any ℤ-homology 3sphere is homology cobordant to a real hyperbolic one. However, it is still unknown whether one can introduce a geometric structure on such a homology cobordism, or a CR-structure on a given real hyperbolic 3-manifold (in particular, a homology sphere) or on a ℤ-homology 3-sphere of plumbing type. We refer to [S, Mat] for recent advances on homology cobordisms, in particular, for results on Floer homology of homology 3-spheres and a new Saveliev’s (presumably, homology cobordism) invariant based on Floer homology.

6. Homeomorphisms induced by group isomorphisms

As another application of the developed methods, we study the following well known problem of geometric realizations of group isomorphisms:

Problem 6.1. Given a type preserving isomorphism φ : G → H of discrete groups G, H ⊂ P U(n, 1), find subsets X_{G}, X_{H} ⊂ \overline{ℍ_{ℂ}^{n}} invariant for the action of groups G and H, respectively, and an equivariant homeomorphism f_{φ} : X_{G} → X_{H} which induces the isomorphism φ. Determine metric properties of f_{φ}, in particular, whether it is either quasisymmetric or quasiconformal.

Such type problems were studied by several authors. In the case of lattices G and H in rank 1 symmetric spaces X, G.Mostow [Mo1] proved in his celebrated rigidity theorem that such isomorphisms φ : G → H can be extended to inner isomorphisms of X, provided that there is no analytic homomorphism of X onto P SL(2, ℝ). For that proof, it was essential to prove that φ can be induced by a quasiconformal homeomorphism of the sphere at infinity ∂X which is the one point compactification of a (nilpotent) Carnot group N (for quasiconformal mappings in Heisenberg and Carnot groups, see [KR, P]).

If geometrically finite groups G, H ⊂ P U(n, 1) have parabolic elements and are neither lattices nor trivial, the only results on geometric realization of their isomorphisms are known in the real hyperbolic space [Tu]. Generally, those methods cannot be used in the complex hyperbolic space due to lack of control over convex hulls (where the convex hull of three points may be 4-dimensional), especially nearby cusps. Another (dynamical) approach due to C.Yue [Yu2, Cor.B] (and the Anosov-Smale stability theorem for hyperbolic flows) can be used only for convex cocompact groups G and H, see [Yu3]. As a first step in solving the general Problem 6.1, we have the following isomorphism theorem [A7]:

Theorem 6.2. Let ϕ : G → H be a type preserving isomorphism of two non-ele-mentary geometrically finite groups G, H ⊂ P U(n, 1). Then there exists a unique equivariant homeomorphism f_{ϕ} : Λ(G) → Λ(H) of their limit sets that induces the isomorphism ϕ. Moreover, if Λ(G) = ∂ℍ_{ℂ}^{n}, the homeomorphism f_{ϕ} is the restriction of a hyperbolic isometry h ∈ P U(n, 1).

Proof. To prove this claim, we consider the Cayley graph K(G, σ) of a group G with a given finite set σ of generators. This is a 1-complex whose vertices are elements of G, and such that two vertices a, b ∈ G are joined by an edge if and only if a = bg^{±1} for some generator g ∈ σ. Let | ∗ | be the word norm on K(G, σ), that is, |g| equals the minimal length of words in the alphabet σ representing a given element g ∈ G. Choosing a function ρ such that ρ(r) = 1/r^{2} for r > 0 and ρ(0) = 1, one can define the length of an edge [a, b] ⊂ K(G, σ) as d_{ρ}(a, b) = min{ρ(|a|), ρ(|b|)}. Considering paths of minimal length in the sense of the function d_{ρ}(a, b), one can extend it to a metric on the Cayley graph K(G, σ). So taking the Cauchy completion \overline{K(G, σ)} of that metric space, we have the definition of the group completion \overline{G} as the compact metric space \overline{K(G, σ)}\K(G, σ), see [Fl]. Up to a Lipschitz equivalence, this definition does not depend on σ. It is also clear that, for a cyclic group ℤ, its completion \overline{ℤ} consists of two points. Nevertheless, for a nilpotent group G with one end, its completion \overline{G} is a one-point set [Fl].

Now we can define a proper equivariant embedding F : K(G, σ) ↪ ℍ_{ℂ}^{n} of the Cayley graph of a given geometrically finite group G ⊂ P U(n, 1). To do that we may assume that the stabilizer of a point, say 0 ∈ ℍ_{ℂ}^{n}, is trivial. Then we set F (g) = g(0) for any vertex g ∈ K(G, σ), and F maps any edge [a, b] ⊂ K(G, σ) to the geodesic segment [a(0), b(0)] ⊂ ℍ_{ℂ}^{n}.

Proposition 6.3. For a geometrically finite discrete group G ⊂ P U(n, 1), there are constants K, K′ > 0 such that the following bounds hold for all elements g ∈ G with |g| ≥ K′:

ln(2|g| − K)^{2} − ln K^{2} ≤ d(0, g(0)) ≤ K|g| .  (6.4)

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} ⊂ ℍ_{ℂ}^{n}. Let C(Λ(G)) ⊂ ℍ_{ℂ}^{n} be the convex hull of the limit set Λ(G) ⊂ ∂ℍ_{ℂ}^{n}, that is the minimal convex subset in ℍ_{ℂ}^{n} whose closure in \overline{ℍ_{ℂ}^{n}} contains Λ(G). Clearly, it is G-invariant, and its quotient C(Λ(G))/G is the minimal convex retract of ℍ_{ℂ}^{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 ℍ_{0}/G. In other words, ℍ_{0} ⊂ C(Λ(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 [AX1, Bow], cf. also [A1, Th. 6.33]. Now, having co-compact action of the group G on the domain ℍ_{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′ = (ξ, v, u) on a “horizontal” horosphere S_{u} = ℂ^{n−1} × ℝ × {u} ⊂ ℍ_{ℂ}^{n}. Then the distances d and d_{0} are as follows [Pr2]:

cosh^{2} \frac{d}{2} = \frac{1}{4u^{2}} (|ξ|^{4} + 4u|ξ|^{2} + 4u^{2} + v^{2}) , d_{0}^{2} = \frac{|ξ|^{2}}{u} + \frac{v^{2}}{4u^{2}} .  (6.5)

This comparison and the basic fact due to Cannon [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, σ) to the group completion \overline{G}. Since the group completion of any parabolic subgroup G_{p} ⊂ 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

Theorem 6.6. For a geometrically finite discrete group G ⊂ P U(n, 1), there is a continuous G-equivariant map Φ_{G} : \overline{G} → Λ(G). Moreover, the map Φ_{G} is bijective everywhere but the set of parabolic fixed points p ∈ Λ(G) whose stabilizers G_{p} ⊂ G have rank one. On this set, the map Φ_{G} is two-to-one.

Now we can finish our proof of Theorem 6.2 by looking at the following diagram of maps:

Λ(G) ⟵^{Φ_{G}} \overline{G} ⟶^{\overline{ϕ}} \overline{H} ⟶^{Φ_{H}} Λ(H) ,

where the homeomorphism \overline{ϕ} is induced by the isomorphism ϕ, and the continuous maps Φ_{G} and Φ_{H} are defined by Theorem 6.6. Namely, one can define a map f_{ϕ} = Φ_{H} \overline{ϕ}Φ_{G}^{−1}. Here the map Φ_{G}^{−1} is the right inverse to Φ_{G}, which exists due to Theorem 6.6. Furthermore, the map Φ_{G}^{−1} is bijective everywhere but the set of parabolic fixed points p ∈ Λ(G) whose stabilizers G_{p} ⊂ G have rank one, where it is 2-to-1. Hence the composition map f_{ϕ} 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 ∈ G must be the attractive fixed point of the loxodromic element ϕ(g) ∈ H (such loxodromic fixed points are dense in the limit set, see [A1]).

The last claim of the Theorem 6.2 directly follows from the Mostow rigidity theorem [Mo1] because a geometrically finite group G ⊂ P U(n, 1) with Λ(G) = ∂ℍ_{ℂ}^{n} is co-finite: vol (ℍ_{ℂ}^{n}/G) < ∞.

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Remark 6.7. Our proof of Theorem 6.2 can be easily extended to the general situation, that is, to construct equivariant homeomorphisms f_{ϕ} : Λ(G) → Λ(H) conjugating the actions (on the limit sets) of isomorphic geometrically finite groups G, H ⊂ Isom X in a (symmetric) space X with pinched negative curvature K, −b^{2} ≤ K ≤ −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 [HI, Th.4.6] which compares the geometry of horospheres S_{u} ⊂ X with that in the spaces of constant curvature −a^{2} and −b^{2}, respectively. It gives, that for all x, y ∈ 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 · d/2) ≤ d_{u} ≤ \frac{2}{b} sinh(b · d/2).

Upon existence of such homeomorphisms f_{φ} inducing given isomorphisms φ of discrete subgroups of P U(n, 1), the Problem 6.1 can be reduced to the questions whether f_{φ} is quasisymmetric with respect to the Carnot-Carathéodory (or Cygan) metric, and whether there exists its G-equivariant extension to a bigger set (to the sphere at infinity ∂X or even to the whole space \overline{ℍ_{ℂ}^{n}}) inducing the isomorphism φ. For convex cocompact groups obtained by nearby representations, this may be seen as a generalization of D.Sullivan stability theorem [Su2], see also [A9].

However, in a deep contrast to the real hyperbolic case, here we have an interesting effect related to possible noncompactness of the boundary ∂M(G) = Ω(G)/G. Namely, even for the simplest case of parabolic cyclic groups G ≅ H ⊂ P U(n, 1), the homeomorphic CR-manifolds ∂M(G) = H^{n}/G and ∂M(H) = H^{n}/H may be not quasiconformally equivalent, see [Min]. In fact, among such Cauchy-Riemannian 3manifolds (homeomorphic to ℝ^{2}×S^{1}), there are exactly two quasiconformal equivalence classes whose representatives have the holonomy groups generated correspondingly by a vertical H-translation by (0, 1) ∈ ℂ × ℝ and a horizontal H-translation by (1, 0) ∈ ℂ × ℝ.

Theorem 7.1 presents a more sophisticated topological deformation {f_{α}}, f_{α} : \overline{ℍ_{ℂ}^{2}} → \overline{ℍ_{ℂ}^{2}}, of a ”complex-Fuchsian” co-finite group G ⊂ P U(1, 1) ⊂ P U(2, 1) to quasi-Fuchsian discrete groups G_{α} = f_{α}Gf_{α}^{−1} ⊂ P U(2, 1). It deforms pure parabolic subgroups in G to subgroups in G_{α} generated by Heisenberg “screw translations”. As we point out, any such G-equivariant conjugations of the groups G and G_{α} 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_{φ}, f_{φ} : Λ(G) → Λ(H). It follows from the next example.

Example 6.7. Let G ⊂ P U(1, 1) ⊂ P U(2, 1) and H ⊂ P O(2, 1) ⊂ P U(2, 1) be two geometrically finite (loxodromic) groups isomorphic to the fundamental group π_{1}(S_{g}) of a compact oriented surface S_{g} of genus g > 1. Then the equivariant homeomorphism f_{φ} : Λ(G) → Λ(H) cannot be homeomorphically extended to the whole sphere ∂ℍ_{ℂ}^{2} ≈ S^{3}.

Proof. The obstruction in this example is topological and is due to the fact that the quotient manifolds M_{1} = ℍ_{ℂ}^{2}/G and M_{2} = ℍ_{ℂ}^{2}/H are not homeomorphic. Namely, these complex surfaces are disk bundles over the Riemann surface S_{g} and have different Toledo invariants: τ(ℍ_{ℂ}^{2}/G) = 2g − 2 and τ(ℍ_{ℂ}^{2}/H) = 0, see [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 [BS]. Moreover due to Goldman [Go1], since the surface S_{p} ⊂ M_{1} is closed, the manifold M_{1} is locally rigid in the sense that every nearby representation G → P U(2, 1) stabilizes a complex geodesic in ℍ_{ℂ}^{2} and is conjugate to a representation G → P U(1, 1) ⊂ P U(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 Teichmüller space of different “quasi-Fuchsian” complex hyperbolic structures.

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7. Deformations of complex hyperbolic and CR-structures: flexibility versus rigidity

Since any real hyperbolic n-manifold can be (totally geodesically) embedded to a complex hyperbolic n-manifold ℍ_{ℂ}^{n}/G, flexibility of the latter ones is evident starting with hyperbolic structures on a Riemann surface of genus g > 1, which form Teichmüller space, a complex analytic (3g − 3)-manifold. Strong rigidity starts in real dimension 3. Namely, due to the Mostow rigidity theorem [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 [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 ×(0, 1) of infinite volume. First such quasi-Fuchsian deformations were given by the author [A2] and, after Thurston’s “Mickey Mouse” example [Th], they were called bendings of M along its totally geodesic hypersurfaces, see also [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 ℍ_{ℝ}^{n}. In particular, the limit set Λ(G) ⊂ ∂ℍ_{ℝ}^{n+1} deforms continuously from a round sphere ∂ℍ_{ℝ}^{n} = S^{n−1} ⊂ S^{n} = ℍ_{ℝ}^{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 [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 [Co2], such manifolds are even super-rigid – analogously to Margulis super-rigidity in higher rank [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 [G1] and its extension for n ≥ 3 [GM], every nearby discrete representation ρ : G → P U(n, 1) of a cocompact lattice G ⊂ P U(n − 1, 1) stabilizes a complex totally geodesic subspace ℍ_{ℂ}^{n−1} in ℍ_{ℂ}^{n}. Thus the limit set Λ(ρG) ⊂ ∂ℍ_{ℂ}^{n} is always a round sphere S^{2n−3}. In higher dimensions n ≥ 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 [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 ⊂ P U(n, 1) preserves a complex totally geodesic hyperspace in ℍ_{ℂ}^{n} is essential for local rigidity of deformations only for co-compact lattices G ⊂ P U(n − 1, 1). This is due to the following our result [ACG]:

Theorem 7.1. Let G ⊂ P U(1, 1) be a co-finite free lattice whose action in ℍ_{ℂ}^{2} is generated by four real involutions (with fixed mutually tangent real circles at infinity). Then there is a continuous family {f_{α}}, −ϵ < α < ϵ, of G-equivariant homeomorphisms in \overline{ℍ_{ℂ}^{2}} which induce non-trivial quasi-Fuchsian (discrete faithful) representations f_{α}^{∗} : G → P U(2, 1). Moreover, for each α ≠ 0, any G-equivariant homeomorphism of \overline{ℍ_{ℂ}^{2}} that induces the representation f_{α}^{∗} cannot be quasiconformal.

This and an Yue’s [Yu2] result on Hausdorff dimension show that there are deformations of a co-finite Fuchsian group G ⊂ P U(1, 1) into quasi-Fuchsian groups G_{α} = f_{α}Gf_{α}^{−1} ⊂ P U(2, 1) with Hausdorff dimension of the limit set Λ(G_{α}) 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 real geodesic surfaces are not rigid, too. Namely, we give a canonical construction of continuous non-trivial quasi-Fuchsian deformations of manifolds M, dim_{ℂ} M = 2, fibered over closed Riemann surfaces, which are the first such deformations of fibrations over compact base (for a noncompact base corresponding to an ideal triangle group G ⊂ P O(2, 1), see [GP2]).

Our construction is inspired by the approach the author used for bending deformations of real hyperbolic (conformal) manifolds along totally geodesic hypersurfaces ([A2, A4]) and by an example of M.Carneiro–N.Gusevskii [Gu] constructing a non-trivial discrete representation of a surface group into P U(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 [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 Kähler and contact structures (correspondingly in ℍ_{ℂ}^{n} and at its infinity, \overline{H_{n}}). Despite that, the constructed complex bending deformations are induced by equivariant homeomorphisms of \overline{ℍ_{ℂ}^{n}}, which are in addition quasiconformal:

Theorem 7.2. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a given (non-elementary) discrete group. Then, for any simple closed geodesic α in the Riemann 2-surface S = H_{ℝ}^{2}/G and a sufficiently small η_{0} > 0, there is a holomorphic family of G-equivariant quasiconformal homeomorphisms F_{η} : \overline{ℍ_{ℂ}^{2}} → \overline{ℍ_{ℂ}^{2}}, −η_{0} < η < η_{0}, which defines the bending (quasi-Fuchsian) deformation B_{α} : (−η_{0}, η_{0}) → R_{0}(G) of the group G along the geodesic α, B_{α}(η) = F_{η}^{∗}.

We notice that deformations of a complex hyperbolic manifold M may depend on many parameters described by the Teichmüller space T (M) of isotopy classes of complex hyperbolic structures on M. One can reduce the study of this space T (M) to studying the variety T (G) of conjugacy classes of discrete faithful representations ρ : G → P U(n, 1) (involving the space D(M) of the developing maps, see [Go2, FG]). Here T (G) = R_{0}(G)/P U(n, 1), and the variety R_{0}(G) ⊂ Hom(G, P U(n, 1)) consists of discrete faithful representations ρ of the group G with infinite co-volume, Vol(ℍ_{ℂ}^{n}/G) = ∞. In particular, our complex bending deformations depend on many independent parameters as it can be shown by applying our construction and Élie Cartan [Car] angular invariant in Cauchy-Riemannian geometry:

Corollary 7.3. Let S_{p} = ℍ_{ℝ}^{2}/G be a closed totally real geodesic surface of genus p > 1 in a given complex hyperbolic surface M = ℍ_{ℂ}^{2}/G, G ⊂ P O(2, 1) ⊂ P U(2, 1). Then there is an embedding π ∘ B : B^{3p−3} ↪ T (M) of a real (3p − 3)-ball into the Teichmüller space of M, defined by bending deformations along disjoint closed geodesics in M and by the projection π : D(M) → T (M) = D(M)/P U(2, 1) in the development space D(M).

Basic Construction (Proof of Theorem 7.2). Now we start with a totally real geodesic surface S = ℍ_{ℝ}^{2}/G in the complex surface M = ℍ_{ℂ}^{2}/G, where G ⊂ P O(2, 1) ⊂ P U(2, 1) is a given discrete group, and fix a simple closed geodesic α on S. We may assume that the loop α is covered by a geodesic A ⊂ ℍ_{ℝ}^{2} ⊂ ℍ_{ℂ}^{2} whose ends at infinity are ∞ and the origin of the Heisenberg group H = ℂ × ℝ, \overline{H} = ∂ℍ_{ℂ}^{2}. Furthermore, using quasiconformal deformations of the Riemann surface S (in the Teichmüller space T (S), that is, by deforming the inclusion G ⊂ P O(2, 1) in P O(2, 1) by bendings along the loop α, see Corollary 3.3 in [A10]), we can assume that the hyperbolic length of α is sufficiently small and the radius of its tubular neighborhood is big enough:

Lemma 7.4. Let g_{α} be a hyperbolic element of a non-elementary discrete group G ⊂ P O(2, 1) ⊂ P U(2, 1) with translation length ℓ along its axis A ⊂ ℍ_{ℝ}^{2}. Then any tubular neighborhood U_{δ}(A) of the axis A of radius δ > 0 is precisely invariant with respect to its stabilizer G_{0} ⊂ G if sinh(ℓ/4) · sinh(δ/2) ≤ 1/2. Furthermore, for sufficiently small ℓ, ℓ < 4δ, the Dirichlet polyhedron D_{z}(G) ⊂ ℍ_{ℂ}^{2} of the group G centered at a point z ∈ A has two sides a and a′ intersecting the axis A and such that g_{α}(a) = a′.

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}) ⊂ ℍ_{ℂ}^{2}, i = 0, 1, 2, centered at a point z ∈ A = (0, ∞), whose intersections with the hyperbolic 2-plane ℍ_{ℝ}^{2} have the shapes indicated in Figures 2-5.

Figure 2. G_{1} ⊂ G = G_{1} ∗_{G_{0}} G_{2} Figure 3. G_{2} ⊂ G = G_{1} ∗_{G_{0}} G_{2}

In particular we have that, except two bisectors 𝔖 and 𝔖′ that are equivalent under the hyperbolic translation g_{α} (which generates the stabilizer G_{0} ⊂ 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) ℝ_{+} and ℝ_{−} of the real circle ℝ × {0} ⊂ ℂ × ℝ = H.

Actually, we may assume that the Heisenberg spheres at infinity of the bisectors 𝔖 and 𝔖′ have radii 1 and r_{0} > 1, correspondingly. Then, for a sufficiently small ϵ, 0 < ϵ << r_{0} −1, the cone neighborhoods C_{+}, C_{−} ⊂ \overline{ℍ_{ℂ}^{2}}\\{∞\} = ℂ×ℝ×[0, +∞) are

Figure 4. G_{1} ⊂ G = G_{1}∗_{G_{0}} Figure 5. G = G_{1}∗_{G_{0}}

correspondingly the cones of the ϵ-neighborhoods of the points (1, 0, 0), (−1, 0, 0) ∈ ℂ × ℝ × [0, +∞) with respect to the Cygan metric ρ_{c} in \overline{ℍ_{ℂ}^{2}}\\{∞\}, see (2.1).

Clearly, we may consider the length ℓ of the geodesic α so small that closures of all equidistant halfspaces in \overline{ℍ_{ℂ}^{2}}\\{∞\} bounded by those bisectors (and whose interiors are disjoint from the Dirichlet polyhedron D_{z}(G)) do not intersect the co-vertical bisector whose infinity is iℝ × ℝ ⊂ ℂ × ℝ. It follows from the fact [Go3, Thm VII.4.0.3] that equidistant half-spaces 𝔖_{1} and 𝔖_{2} in ℍ_{ℂ}^{2} are disjoint if and only if the half-planes 𝔖_{1} ∩ ℍ_{ℝ}^{2} and 𝔖_{2} ∩ ℍ_{ℝ}^{2} are disjoint, see Figures 2-5.

Now we are ready to define a quasiconformal bending deformation of the group G along the geodesic A, which defines a bending deformation of the complex surface M = ℍ_{ℂ}^{2}/G along the given closed geodesic α ⊂ S ⊂ M.

We specify numbers η and ζ such that 0 < ζ < π/2, 0 ≤ η < π − 2ζ and the intersection C_{+} ∩ (ℂ × {0}) is contained in the angle {z ∈ ℂ : | arg z| ≤ ζ}. Then we define a bending homeomorphism ϕ = ϕ_{η,ζ} : ℂ → ℂ which bends the real axis ℝ ⊂ ℂ at the origin by the angle η, see Fig. 6:

ϕ_{η,ζ}(z) = \{ \begin{matrix}z & if | arg z| ≥ π − ζ \\ z · exp(iη) & if | arg z| ≤ ζ \\ z · exp(iη(1 − (arg z − ζ)/(π − 2ζ))) & if ζ < arg z < π − ζ \\ z · exp(iη(1 + (arg z + ζ)/(π − 2ζ))) & if ζ − π < arg z < −ζ . \end{matrix} (7.5)

Figure 6

For negative η, 2ζ − π < η < 0, we set ϕ_{η,ζ}(z) = \overline{ϕ_{−η,ζ}(\overline{z})}. Clearly, ϕ_{η,ζ} is quasiconformal with respect to the Cygan norm (2.1) and is an isometry in the ζ-cone neighborhood of the real axis ℝ because its linear distortion is given by

K(ϕ_{η,ζ}, z) = \{ \begin{matrix}1 & if | arg z| ≥ π − ζ \\ 1 & if | arg z| ≤ ζ \\ (π − 2ζ)/(π − 2ζ − η) & if ζ < arg z < π − ζ \\ (π − 2ζ + η)/(π − 2ζ) & if ζ − π < arg z < −ζ . \end{matrix}  (7.6)

Foliating the punctured Heisenberg group H\{0} by Heisenberg spheres S(0, r) of radii r > 0, we can extend the bending homeomorphism ϕ_{η,ζ} to an elementary bending homeomorphism φ = φ_{η,ζ} : H → H, φ(0) = 0, φ(∞) = ∞, of the whole sphere S^{3} = \overline{H} at infinity.

Namely, for the “dihedral angles” W_{+}, W_{−} ⊂ H with the common vertical axis {0} × ℝ and which are foliated by arcs of real circles connecting points (0, v) and (0, −v) on the vertical axis and intersecting the the ζ-cone neighborhoods of infinite rays ℝ_{+}, ℝ_{−} ⊂ ℂ, correspondingly, the restrictions φ|_{W_{−}} and φ|_{W_{+}} of the bending homeomorphism φ = φ_{η,ζ} are correspondingly the identity and the unitary rotation U_{η} ∈ P U(2, 1) by angle η about the vertical axis {0} × ℝ ⊂ H, see also [A10, (4.4)]. Then it follows from (7.6) that φ_{η,ζ} is a G_{0}-equivariant quasiconformal homeomorphism in \overline{H}.

We can naturally extend the foliation of the punctured Heisenberg group H\{0} by Heisenberg spheres S(0, r) to a foliation of the hyperbolic space ℍ_{ℂ}^{2} by bisectors 𝔖_{r} having those S(0, r) as the spheres at infinity. It is well known (see [M2]) that each bisector 𝔖_{r} contains a geodesic γ_{r} which connects points (0, −r^{2}) and (0, r^{2}) of the Heisenberg group H at infinity, and furthermore 𝔖_{r} fibers over γ_{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 ℍ_{ℂ}^{2} and bisectors 𝔖_{r}, we extend the elementary bending homeomorphism φ_{η,ζ} : \overline{H} → \overline{H} at infinity to an elementary bending homeomorphism Φ_{η,ζ} : \overline{ℍ_{ℂ}^{2}} → \overline{ℍ_{ℂ}^{2}}. Namely, the map Φ_{η,ζ} preserves each of bisectors 𝔖_{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 ∞ of the Heisenberg group H at infinity. We complete our extension Φ_{η,ζ} by defining its restriction to a given (invariant) complex geodesic fiber Y with the fixed point y ∈ Y. This map is obtained by radiating the circle homeomorphism φ_{η,ζ}|_{∂Y} to the whole (Poincaré) hyperbolic 2-plane Y along geodesic rays [y, ∞) ⊂ Y, so that it preserves circles in Y centered at y and bends (at y, by the angle η) the geodesic in Y connecting the central points of the corresponding arcs of the complex circle ∂Y, see Fig.6.

Due to the construction, the elementary bending (quasiconformal) homeomorphism Φ_{η,ζ} commutes with elements of the cyclic loxodromic group G_{0} ⊂ G. Another most important property of the homeomorphism Φ_{η,ζ} 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} ⊂ G, and 𝔖^{+} ⊂ ℍ_{ℂ}^{2} be a “half-space” disjoint from D_{z}(G) and bounded by a bisector 𝔖 ⊂ ℍ_{ℂ}^{2} which is different from bisectors 𝔖_{r}, r > 0, and contains a side 𝔰 of the polyhedron D_{z}(G). Then there is an open neighborhood U(\overline{𝔖^{+}}) ⊂ \overline{ℍ_{ℂ}^{2}} such that the restriction of the elementary bending homeomorphism Φ_{η,ζ} to it either is the identity or coincides with the unitary rotation U_{η} ⊂ P U(2, 1) by the angle η about the “vertical” complex geodesic (containing the vertical axis {0} × ℝ ⊂ H at infinity).

The above properties of quasiconformal homeomorphism Φ = Φ_{η,ζ} show that the image D_{η} = Φ_{η,ζ}(D_{z}(G)) is a polyhedron in ℍ_{ℂ}^{2} bounded by bisectors. Furthermore, there is a natural identification of its sides induced by Φ_{η,ζ}. Namely, the pairs of sides preserved by Φ are identified by the original generators of the group G_{1} ⊂ G. For other sides 𝔰_{η} of D_{η}, which are images of corresponding sides 𝔰 ⊂ D_{z}(G) under the unitary rotation U_{η}, 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 ∈ G_{2} of D_{z}(G)-sides to the hyperbolic isometries U_{η}gU_{η}^{−1} ∈ P U(2, 1). In the case of HNNextension, G = G_{1}∗_{G_{0}} = ⟨G_{1}, g_{2}⟩, we change the original side pairing g_{2} ∈ G of D_{z}(G)-sides to the hyperbolic isometry U_{η}g_{2} ∈ P U(2, 1). In other words, we define deformed groups G_{η} ⊂ P U(2, 1) correspondingly as

G_{η} = G_{1} ∗_{G_{0}} U_{η}G_{2}U_{η}^{−1} or G_{η} = ⟨G_{1}, U_{η}g_{2}⟩ = G_{1} ∗_{G_{0}} .  (7.7)

This shows that the family of representations G → G_{η} ⊂ P U(2, 1) does not depend on angles ζ and holomorphically depends on the angle parameter η. Let us also observe that, for small enough angles η, the behavior of neighboring polyhedra g′(D_{η}), g′ ∈ G_{η} is the same as of those g(D_{z}(G)), g ∈ G, around the Dirichlet fundamental polyhedron D_{z}(G). This is because the new polyhedron D_{η} ⊂ ℍ_{ℂ}^{2} has isometrically the same (tesselations of) neighborhoods of its side-intersections as D_{z}(G) had. This implies that the polyhedra g′(D_{η}), g′ ∈ G_{η}, form a tesselation of ℍ_{ℂ}^{2} (with non-overlapping interiors). Hence the deformed group G_{η} ⊂ P U(2, 1) is a discrete group, and D_{η} is its fundamental polyhedron bounded by bisectors.

Using G-compatibility of the restriction of the elementary bending homeomorphism Φ = Φ_{η,ζ} to the closure \overline{D_{z}(G)} ⊂ \overline{ℍ_{ℂ}^{2}}, we equivariantly extend it from the polyhedron \overline{D_{z}(G)} to the whole space ℍ_{ℂ}^{2} ∪ Ω(G) accordingly to the G-action. In fact, in terms of the natural isomorphism χ : G → G_{η} which is identical on the subgroup G_{1} ⊂ G, we can write the obtained G-equivariant homeomorphism F = F_{η} : \overline{ℍ_{ℂ}^{2}}\Λ(G) → \overline{ℍ_{ℂ}^{2}}\Λ(G_{η}) in the following form:

\begin{aligned}F_{η}(x) & = Φ_{η}(x) for x ∈ \overline{D_{z}(G)}, \\ F_{η} ∘ g(x) & = g_{η} ∘ F_{η}(x) for x ∈ \overline{ℍ_{ℂ}^{2}}\Λ(G), g ∈ G, g_{η} = χ(g) ∈ G_{η} . \end{aligned} (7.8)

Due to quasiconformality of Φ_{η}, the extended G-equivariant homeomorphism F_{η} is quasiconformal. Furthermore, its extension by continuity to the limit (real) circle Λ(G) coincides with the canonical equivariant homeomorphism f_{χ} : Λ(G) → Λ(G_{η}) given by the isomorphism Theorem 6.2. Hence we have a G-equivariant quasiconformal self-homeomorphism of the whole space \overline{ℍ_{ℂ}^{2}}, which we denote as before by F_{η}.

The family of G-equivariant quasiconformal homeomorphisms F_{η} induces representations F_{η}^{∗} : G → G_{η} = F_{η}G_{2}F_{η}^{−1}, η ∈ (−η_{0}, η_{0}). In other words, we have a curve B : (−η_{0}, η_{0}) → R_{0}(G) in the variety R_{0}(G) of faithful discrete representations of G into P U(2, 1), which covers a nontrivial curve in the Teichmüller space T (G) represented by conjugacy classes [B(η)] = [F_{η}^{∗}]. We call the constructed deformation B the bending deformation of a given lattice G ⊂ P O(2, 1) ⊂ P U(2, 1) along a bending geodesic A ⊂ ℍ_{ℂ}^{2} with loxodromic stabilizer G_{0} ⊂ G. In terms of manifolds, B is the bending deformation of a given complex surface M = ℍ_{ℂ}^{2}/G homotopy equivalent to its totally real geodesic surface S_{g} ⊂ M, along a given simple geodesic α.

□

Remark 7.9. It follows from the above construction of the bending homeomorphism F_{η,ζ}, that the deformed complex hyperbolic surface M_{η} = ℍ_{ℂ}^{2}/G_{η} fibers over the pleated hyperbolic surface S_{η} = F_{η}(ℍ_{ℝ}^{2})/G_{η} (with the closed geodesic α 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 η 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} ⊂ G and the totally real geodesic planes that represent fibers of the original fibration in M = ℍ_{ℂ}^{2}/G.

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 [B(η)] ≠ [B(η′)] for any η ≠ η′.

The non-triviality of our deformation follows directly from (7.7), cf. [A9]. Namely, the restrictions ρ_{η}|_{G_{1}} of bending representations to a non-elementary subgroup G_{1} ⊂ G (in general, to a “real” subgroup G_{r} ⊂ G corresponding to a totally real geodesic piece in the homotopy equivalent surface S ⋍ M) are identical. So if the deformation 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} ⊂ G and pointwise fix the totally real geodesic plane ℍ_{ℝ}^{2}. This contradicts to the fact that the limit set of any deformed group G_{η}, η ≠ 0, does not belong to the real circle containing the limit Cantor set Λ(G_{r}).

The injectivity of the map B can be obtained by using Élie Cartan [Car] angular invariant 𝔸(x), −π/2 ≤ 𝔸(x) ≤ π/2, for a triple x = (x^{0}, x^{1}, x^{2}) of points in ∂ℍ_{ℂ}^{2}. It is known (see [Go3]) that, for two triples x and y, 𝔸(x) = 𝔸(y) if and only if there exists g ∈ P U(2, 1) such that y = g(x); furthermore, such a g is unique provided that 𝔸(x) is neither zero nor ±π/2. Here 𝔸(x) = 0 if and only if x^{0}, x^{1} and x^{2} lie on an ℝ-circle, and 𝔸(x) = ±π/2 if and only if x^{0}, x^{1} and x^{2} lie on a chain (ℂ-circle).

Namely, let g_{2} ∈ G\G_{1} be a generator of the group G in (4.5) whose fixed point x^{2} ∈ Λ(G) lies in ℝ_{+} × {0} ⊂ H, and x_{η}^{2} ∈ Λ(G_{η}) the corresponding fixed point of the element χ_{η}(g_{2}) ∈ G_{η} under the free-product isomorphism χ_{η} : G → G_{η}. Due to our construction, one can see that the orbit γ(x_{η}^{2}), γ ∈ G_{0}, under the loxodromic (dilation) subgroup G_{0} ⊂ G ∩ G_{η} approximates the origin along a ray (0, ∞) which has a non-zero angle η with the ray ℝ_{−} × {0} ⊂ H. The latter ray also contains an orbit γ(x^{1}), γ ∈ 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_{η} = (x^{1}, 0, x_{η}^{2}) of points which lie correspondingly in the limit sets Λ(G) and Λ(G_{η}), we have that 𝔸(x) = 0 and 𝔸(x_{η}) ≠ 0, ±π/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 η and η′, we have triples x_{η} and x_{η′} with different (non-trivial) Cartan angular invariants 𝔸(x_{η}) ≠ 𝔸(x_{η′}). Hence Λ(G_{η}) and Λ(G_{η′}) are not P U(2, 1)-equivalent.

□

One can apply the above proof to a general situation of bending deformations of a complex hyperbolic surface M = ℍ_{ℂ}^{2}/G whose holonomy group G ⊂ P U(2, 1) has a non-elementary subgroup G_{r} preserving a totally real geodesic plane ℍ_{ℝ}^{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 [GKL] with a given Toledo invariant lie in this class. So we immediately have:

Corollary 7.10. Let M = ℍ_{ℂ}^{2}/G be a complex hyperbolic surface with embedded totally real geodesic surface S_{r} ⊂ M with geodesic boundary, and B : (−η, η) → D(M) be the bending deformation of M along a simple closed geodesic α ⊂ S_{r}. Then the map π ∘ B : (−η, η) → T (M) = D(M)/P U(2, 1) is a smooth embedding provided that the limit set Λ(G) of the holonomy group G does not belong to the G-orbit of the real circle S_{ℝ}^{1} and the chain S_{ℂ}^{1}, where the latter is the infinity of the complex geodesic containing a lift α̃ ⊂ ℍ_{ℂ}^{2} of the closed geodesic α, and the former one contains the limit set of the holonomy group G_{r} ⊂ G of the geodesic surface S_{r}.

□

As an application of the constructed bending deformations, we answer a well known question about cusp groups on the boundary of the Teichmüller space T (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 [AG]:

Theorem 7.11. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a non-elementary discrete group S_{p} of genus p ≥ 2. Then, for any simple closed geodesic α in the Riemann surface S = H_{ℝ}^{2}/G, there is a continuous deformation ρ_{t} = f_{t}^{∗} induced by G-equivariant quasiconformal homeomorphisms f_{t} : \overline{ℍ_{ℂ}^{2}} → \overline{ℍ_{ℂ}^{2}} whose limit representation ρ_{∞} corresponds to a boundary cusp point of the Teichmüller space T (G), that is, the boundary group ρ_{∞}(G) has an accidental parabolic element ρ_{∞}(g_{α}) where g_{α} ∈ G represents the geodesic α ⊂ S.

We note that, due to our construction of such continuous quasiconformal deformations in [AG], they are independent if the corresponding geodesics α_{i} ⊂ S_{p} are disjoint. It implies the existence of a boundary group in ∂T (G) with “maximal” number of non-conjugate accidental parabolic subgroups:

Corollary 7.12. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a uniform lattice isomorphic to the fundamental group of a closed surface S_{p} of genus p ≥ 2. Then there is a continuous deformation R : ℝ^{3p−3} → T (G) whose boundary group G_{∞} = R(∞)(G) has (3p − 3) non-conjugate accidental parabolic subgroups.

Finally, we mention another aspect of the intrigue Problem 4.12 on geometrical finiteness of complex hyperbolic surfaces (see [AX1, AX2]) for which it may perhaps be possible to apply our complex bending deformations:

Problem. Construct a geometrically infinite (finitely generated) discrete group G ⊂ P U(2, 1) whose limit set is the whole sphere at infinity, Λ(G) = ∂ℍ_{ℂ}^{2} = \overline{H}, and which is the limit of convex cocompact groups G_{i} ⊂ P U(2, 1) from the Teichmüller space T (Γ) of a convex cocompact group Γ ⊂ P U(2, 1). Is that possible for a Schottky group Γ?

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Department of Mathematics, University of Oklahoma, Norman, OK 73019 E-mail address: Apanasov@ou.edu

Sobolev Inst. of Mathematics, Russian Acad. Sci., Novosibirsk, Russia 630090

Mathematical Sciences Research Institute, Berkeley, CA 94720-5070