QUASICONFORMALITY AND GEOMETRICAL

FINITENESS IN CARNOT–CARATHÉODORY

AND NEGATIVELY CURVED SPACES

Boris Apanasov

ABSTRACT. The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot–Carathéodory structures. Especially, the most interesting are complex hyperbolic manifolds with Cauchy–Riemannian structure at infinity, which occupy a distinguished niche and whose properties make them surprisingly different from real hyperbolic ones.

1. Introduction

The paper sketches a recent progress and formulates several open problems in studying quasiconformal and quasisymmetric homeomorphisms as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot–Carathéodory structures in the sense of M. Gromov [Gr]. Here, complex hyperbolic and Cauchy–Riemannian manifolds occupy a distinguished niche. First, due to their complex analytic nature, a broad spectrum of techniques can contribute to the study, and already obtained results show surprising differences between geometry and topology of noncompact complex and real hyperbolic manifolds, see [BS, BuM, EMM, GoM, KR1-2]. Second, these Kähler manifolds are the most known manifolds with variable pinched negative curvature [BGS, B, Mg1, MaG]. Finally, for complex analytic surfaces, one can apply Seiberg-Witten invariants, decomposition of 4-manifolds along homology 3spheres, Floer homology and new (homology) cobordism invariants [LB, A11, Sa1-Sa4].

There are three main themes united by using of general Thurston’s idea of geometric uniformization of low dimensional manifolds provided some canonical decomposition of them into geometric pieces. First, basic geometric block-manifolds of negative (variable) curvature which possess geometrical finiteness and may however have infinite volume. Second, the interaction between the negatively curved

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 57,55,53,51.

Key words and phrases. negatively curved manifolds; nilpotent, Carnot and Heisenberg groups; discrete isometry groups; geometrical finiteness; complex hyperbolic manifolds; Cauchy–Riemannian structure; quasiconformal and quasisymmetric mappings; varieties of representations; deformations.

Supported in part by the NSF. Research at MSRI is supported in part by NSF grant DMS-9022140.

geometry of such blocks and the induced Carnot–Carathéodory structure of their boundary manifolds at infinity, especially the interaction between Kähler geometry of geometrically finite complex hyperbolic manifolds and Cauchy–Riemannian structure of their boundaries modeled on the Heisenberg group (which is a particular case of Carnot groups). And finally, deformations of negatively curved manifolds inducing automorphisms of boundaries at infinity which preserves their natural contact structures, together with their metrical (quasiconformal or quasisymmetric) properties.

Our study of geometrical finiteness and homeomorphisms in spaces with negative (variable) curvature exploits a new structural theorem about isometric actions of discrete groups on nilpotent Lie groups (in particular on the Heisenberg group H_{n}). The problems there have a unique appeal both for the amount of similarity with the model situation of interactions between real hyperbolic geometrically finite manifolds, their boundaries with natural conformal structures and their quasiconformal deformations (see [A1, A11]), and for the interesting ways in which the similarity breaks down.

Besides geometrical finiteness in negatively curved spaces, one of inspiring ideas of our study is a well known theorem of D.Sullivan, which gave rise to many important results in geometry and topology of manifolds and theory of quasiconformal mappings. It states, see [Su1, TV], that homeomorphisms of quasiconformal n-manifolds, n ≠ 4, can be approximated by quasiconformal ones, where one of important classes is the class of conformal manifolds. Here by the quasiconformality one means the boundedness of distortion with respect to the Euclidean metric. This result rises questions of approximation in the quasiconformal category in another negatively curved geometries and the corresponding sub-Riemannian Carnot–Carathéodory structures which appear at infinity of those geometries as well. We mention here a recent important development by M.Gromov on Carnot– Carathéodory spaces [Gr], where he shown that continues mappings can be approximated by mappings that are Lipschitz with respect to the Carnot–Carathéodory metric. Another important achievement is a recent result by G.Margulis and G.Mostow [MM] that quasiconformal mappings of Carnot–Carathéodory spaces are a.e. differentiable and preserve their contact structures.

Another set of problems is related to the stability theorem proven by D.Sullivan [Su3] for planar Kleinian groups (see also [A11, MaG]). It raises questions on deformations of discrete isometry groups in spaces of negative (variable) curvature, varieties of their representations (Teichmüller spaces) and their boundaries, and quasiconformal/quasisymmetric homeomorphisms induced by isomorphisms of geometrically finite groups (in conformal category, it is known as Tukia isomorphism theorems [Tu]).

2. Complex hyperbolic and Heisenberg manifolds

The natural class of manifolds where the above problems can be considered is the class of geometrically finite locally homogeneous manifolds. That is why we start with studying such manifolds and their boundaries at infinity, especially complex hyperbolic manifolds and Cauchy–Riemannian and Heisenberg manifolds at their infinity.

We recall some facts concerning the link between nilpotent geometry of the Heisenberg group and the Kähler geometry of the complex hyperbolic space (see [Go, GP1, KR1-2]). Let 𝔹_{ℂ}^{n} ⊂ ℂ^{n} be the unit complex ball equipped with the Bergman metric d which turns the ball into the complex hyperbolic n-space ℍ_{ℂ}^{n} whose sectional curvature is between -1 and -1/4. The automorphism group of ℍ_{ℂ}^{n} ≅ 𝔹_{ℂ}^{n} is the projective unitary group P U(n, 1) whose elements g ∈ P U(n, 1) are biholomorphisms of 𝔹_{ℂ}^{n} 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. Since P U(n, 1) can be embedded in a linear group (for details, see [AX1]), any finitely generated group G ⊂ P U(n, 1) is residually finite and has a finite index torsion free subgroup.

For any point ∞ ∈ ∂𝔹_{ℂ}^{n}, one can identify \overline{𝔹_{ℂ}^{n}}\\{∞\} with the closure of the Siegel domain 𝔖_{n}, which is conveniently represented in horospherical coordinates as ℂ^{n−1}×ℝ×[0, ∞) ([GP1]). Then “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, and the inverse of (ξ, v) is (ξ, v)^{−1} = (−ξ, −v). The Heisenberg group H_{n} isometrically acts 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) .

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, ∞) .

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. Geometrical finiteness in negative curvature

Our main assumption on a negatively curved n-manifold M is the geometrical finiteness condition on its fundamental group G ⊂ Isom X acting by isometries on a simply connected space X = M̃, which in particular implies (see below) that the discrete group G is finitely generated.

A subgroup G ⊂ Isom X is called discrete if it is a discrete subset of Isom X. The limit set Λ(G) ⊂ ∂X of a discrete group G is the set of accumulation points of (any) orbit G(y), y ∈ X. The complement of Λ(G) in ∂X 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 (c.f. [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 below simplify the situation.

Due to the absence of totally geodesic hypersurfaces in a space X 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, one can use another definition of geometrically finite groups G ⊂ Isom X as those ones whose limit sets Λ(G) ⊂ ∂X 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 [Md, Th, A3, A1]) but also in spaces with variable pinched negative curvature [Bo].

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) in the complex hyperbolic space X = ℍ_{ℂ}^{n} are fundamental polyhedra for a discrete subgroup G ⊂ P U(n, 1). They are bounded by bisectors in a complicated way (see [ Mo2, GP1]), and the bisectors are not totally geodesic hypersurfaces. For discrete parabolic groups G ⊂ Isom X, one may expect that the Dirichlet polyhedron D_{y}(G) centered at a point y lying in a G-invariant subspace has finitely many sides. It is true for real hyperbolic spaces [A1] as well as for cyclic and dihedral parabolic groups in complex hyperbolic spaces X = ℍ_{ℂ}^{n}. 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 to [GP1], such 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 3.1. 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.

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]:

Theorem 3.2. Let G ⊂ P U(2, 1) be a discrete parabolic group conjugate to the following subgroup Γ of the Heisenberg group H_{2} = ℂ × ℝ:

Γ = {(m, n) ∈ ℂ × ℝ:m, n ∈ ℤ} .

Then any Dirichlet polyhedron D_{y}(G) centered at an arbitrary point y ∈ ℍ_{ℂ}^{2} has infinitely many sides.

Despite this, we are providing [AX1] 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 construction 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}, which may solve the following problem:

Problem 3.3. Given geometrically finite group G ⊂ Isom X in a negatively curved space X, is there any finitely sided fundamental polyhedron P (G) ⊂ X?

As another tool for attacking this problem, one can use the following structural theorem for discrete groups acting on nilpotent Lie groups (lying at infinity of X), in particular on the Heisenberg group H_{n}, H_{n} ∪ {∞} = ∂ℍ_{ℂ}^{n} (see [AX1- AX3]):

Theorem 3.4. 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. Then 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.

Here we remark that

(1) Compactness of C is an essential condition because of Margulis [Mg2] construction of nonabelian free discrete subgroups Γ of R^{3} ⋊ GL(3, R).

(2) This theorem generalizes a result by L.Auslander [Au] who claimed those properties not for whole group Γ but only for some finite index subgroup. We also remark that an extension in [AX1] of Wolf’s argument [Wo] to the complex hyperbolic case (based on Margulis Lemma [Mg1, BGS] and geometry of H_{n}) does not work in general nilpotent groups. A different algebraic proof of Theorem 3.4 can be found in [AX2, AX3].

This result allows one to study

Problem 3.5. Investigate parabolic cusp ends of negatively curved manifolds.

To solve this problem in the complex hyperbolic case, we use the above Theorem 3.4 to define standard cusp neighborhoods of parabolic cusps. This allows us to prove the following results about cusp ends of complex hyperbolic manifolds (or, equivalently, about the structure of Heisenberg manifolds).

Namely, suppose a point p ∈ ∂ℍ_{ℂ}^{n} is fixed by some parabolic element of the group G, and G_{p} is the stabilizer of p in G. Conjugating the group G by an element h ∈ P U(n, 1) , h(p) = ∞, we have that G_{∞} ⊂ H(n). In particular, if p is the origin in H_{n}, we can take such h as the Heisenberg inversion I in the hyperchain in H_{n}, which preserves the unit Heisenberg sphere S_{c}(0, 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 .  (3.6)

Then, due to Theorem 3.4, there exists a connected Lie subgroup H_{∞} ⊆ H_{n} preserved by G_{∞}.

Definition. 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) = {(ξ, v):ρ_{c}(p, (ξ, v)) = 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 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 [AX1, §5.4]. Applying Theorem 3.4 and [Bo], we have:

Lemma 3.7. Let p ∈ ∂ℍ_{ℂ}^{n} be a parabolic fixed point of a discrete group G ⊂ P U(n, 1). Then p is a cusp point if and only if (Λ(G)\{p})/G_{p} is compact.

This fact and [Bo] allows us to use another equivalent definitions of geometrical finiteness. In particular, a group G ⊂ 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 as U_{p,r}/G_{p} ≈ (S_{p,r_{0}}/G_{p}) × (0, 1], where

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

Using the above description of discrete group actions on a nilpotent group (Theorem 3.4), one can study Carnot–Carathéodory manifolds whose fundamental groups act at infinity of a negatively curved space X as discrete parabolic groups. In particular, we establish fiber bundle structures on Heisenberg manifolds which have the form H_{n}/G where G is a discrete group freely acting on H_{n} by isometries, i.e. a torsion free discrete subgroup of H(n) = H_{n} ⋊ U(n − 1), see [AX1]:

Theorem 3.8. 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.

It gives, in addition to finiteness of generators of geometrically finite groups established in [Bo], the following important finiteness:

Corollary 3.9. The fundamental groups of Heisenberg manifolds and geometrically finite complex hyperbolic manifolds are finitely presented.

Due to Theorem 3.4, any Heisenberg manifold N = H_{n}/Γ is the vector bundle H_{n}/Γ → H_{Γ}/Γ where H_{Γ} ⊂ H_{n} is a minimal Γ-invariant subspace. As simple examples in [AX1] show, such vector bundles are non-trivial in general. However, up to finite coverings, they are trivial [AX1]:

Theorem 3.10. 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.

Such finite covering property holds not only for Heisenberg manifolds alone but for geometrically finite complex hyperbolic manifolds, too:

Theorem 3.11. 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.

It seems this property holds for discrete isometry groups in general negatively curved spaces because our proof of Theorem 3.11 in [AX1] is based on the residual finiteness of geometrically finite subgroups G ⊂ P U(n, 1) and the following two lemmas (the second of which generalizes a result for finite volume real hyperbolic manifolds ([ AF]).

Lemma 3.12. 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 4.1). Then G acts on H_{G} by translations if G is either Abelian or 2-step nilpotent.

Lemma 3.13. 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 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) is geometrically finite. This and Goldman’s local rigidity theorem for uniform lattices G ⊂ U(1, 1) ⊂ P U(2, 1) (see [GM]) suggest the following intrigue question:

Problem 3.14. Are all finitely generated discrete groups G ⊂ P U(2, 1) with nonempty discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{2} geometrically finite?

To solve this problem, one can try to decompose a given finitely generated discrete group G ⊂ P U(2, 1) with non-empty discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{2} into free amalgamated products of elementary groups. More arguments for an affirmative solution of this problem are given by the following two facts. First, due to Chen-Greenberg [CG], all pure loxodromic subgroups G ⊂ P U(2, 1) are discrete. The second fact is due to the trace classification of projective transformations [Go, VI.2]. Namely, in contrast to Kleinian groups on the plane, the subset of groups in a deformation space of a pure loxodromic group G ⊂ P U(2, 1) which have accidental parabolic elements may have real codimension 1.

4. The boundary at infinity of negatively curved manifolds

Another set of problems is related to geometry and topology of Carnot–Carathéodory manifolds which are the boundaries at infinity of negatively curved noncompact manifolds.

In a sharp contrast to the real hyperbolic case, for a compact complex manifold M(G) = (ℍ_{ℂ}^{n} ∪ Ω(G))/G, an application of Kohn-Rossi analytic extension theorem shows that the boundary of this manifold M(G) is connected, and the limit set Λ(G) is in some sense small (see [EMM] and, for quaternionic and Cayley hyperbolic manifolds, [C1]). Moreover, according to a recent result of D.Burns (based on an uniformization theorem [ BuM] for isolated ends of complex analytic spaces), the same claim about connectedness of the boundary ∂M(G) still holds if only a boundary component is compact.

However, if ∂M(G) has no compact components, and there is no finiteness condition on the holonomy group of M(G), our algorithmical construction in [AX1] shows that the situation is completely different:

Theorem 4.1. For any integers k, k_{0}, k ≥ k_{0} ≥ 0, and n ≥ 2, 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 (n−1)-manifolds, ∂_{∞}M = N_{1}∪⋯∪N_{k}. Moreover, for each boundary component N_{j}, j ≤ k_{0}, the inclusion i_{j} : N_{j} ⊂ M(G) induces a homotopy equivalence of N_{j} to M(G).

The construction in the proof of this theorem firstly provides discrete groups G ⊂ P U(n, 1) with two connected components of the discontinuity set Ω(G) ⊂ ∂ℍ_{ℂ}^{n}. Here we essentially use properties of Heisenberg inversions (3.6) in hyperchains in ℍ_{ℂ}^{n}. Then we apply an idea due to A.Tetenov [Te, KAG] in order to construct groups G with any given number k of topological G-invariant balls Ω_{i} ⊂ Ω(G) with common boundary ∂Ω_{i} = Λ(G). The groups we construct in the proof are all however infinitely generated. The finitely generated case has a close relation to Problem 3.14:

Problem 4.2. How many boundary components are there in a complex manifold M(G) with finitely generated fundamental group G ⊂ P U(n, 1)?

Toward this problem, we can show that the situation described in Theorem 4.1 is impossible if the complex hyperbolic manifold M is geometrically finite. Namely, if the 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) homotopy equivalent to M(G), and M(G) can be easily reconstructed from M_{c} by gluing up a finite number of standard open “Heisenberg collars”, see [AX1]:

Theorem 4.3. 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−1−k} 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).

We refer the reader to [AX1] for more precise formulation and proof of this cobordism theorem.

This result allows one to study complex surfaces and Cauchy–Riemannian 3-manifolds at their infinity by using decomposition of such 4-manifolds along homology 3-spheres and applying the gauge theory together with homology (cobordism) invariants. We mention that, due to Milnor [Ml], all Seifert homology 3spheres can be seen as the boundaries at infinity of (geometrically finite) complex hyperbolic 2-orbifolds. Along this line, one can also investigate the following question (which is also related to Problem 5.4):

Problem 4.4. Are there Cauchy–Riemannian structures on homology 3-spheres of plumbing type or on real hyperbolic homology 3-spheres?

One can be especially interested in this question for homology spheres obtained by splicing of two Seifert homology spheres along their singular fibers, see [Sa4]. Another interesting fact (due to Livingston-Myers construction [My]) is that any homology 3-sphere is homology cobordant to a hyperbolic one. On the other hand, as it 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 complex surface. We mention here Yau’s uniformization theorem [Y] which implies that every smooth complex projective 2-surface 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.

We refer to [Sa1-Sa4] and [Mat] for recent advances in this direction, in particular, for results on Floer homology of homology 3-spheres and a new Saveliev’s (presumably, homology cobordism) invariant based on Floer homology.

5. Homeomorphisms induced by group isomorphisms

The main problem we are concerned in this section is about geometric realizations of isomorphisms of geometrically finite discrete groups G, H ⊂ Isom X of isometries of a negatively curved space X:

Problem 5.1. Given an isomorphism φ: G → H of geometrically finite discrete groups G, H ⊂ Isom X, find subsets X_{G}, X_{H} ⊂ \overline{X} 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 with respect to the given negatively curved metric d in X (or the induced sub-Riemannian structure on the Carnot– Carathéodory space at infinity Y = \overline{X}\\{∞\}).

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 \overline{X} which is a one point compactification of a Carnot group N. Quasiconformal mappings between general Carnot groups have been studied by P.Pansu [P]. For the case of complex hyperbolic spaces X = ℍ_{ℂ}^{n} and the Heisenberg groups at their infinity, foundations of the theory of quasiconformal mappings has been made by A.Koranyi and M.Reimann [KR1, KR2].

The essential component of such rigidity results is the fact that quasiconformal mappings of Carnot–Carathéodory spaces are almost everywhere differentiable and preserve the contact structures of these spaces (horizontal vector fields). This fact is due to Pansu [P] in the case of graded nilpotent groups. In the case of general Carnot–Carathéodory spaces determined by “horizontal” subbundles of their tangent bundles, this result has been recently proven by G.Margulis and G.Mostow [MM]. The second important ingredient for such rigidity is the ACL-property on lines for quasiconformal mappings, with respect to the Carnot–Carathéodory metric at infinity \overline{X}\\{∞\}, see [Mo3, KR2, Vo1-2, VG].

If the groups G, H ⊂ Isom X are neither lattices nor trivial, the only results on geometric realization of their isomorphisms are known for real hyperbolic spaces X (of constant negative curvature). In dimension dim X ≥ 3 they are due to P.Tukia [Tu] and use a natural “type-preserving” condition on the isomorphism φ, that parabolic elements are carried out only to parabolic ones.

In the case of variable negative curvature, it is problematic to use geometric properties of convex hulls which provide a powerful tool for studying metric properties of G-equivariant mappings in the real hyperbolic space. However, as a first step in solving the above Problem 5.1, we can analyze the metric of X and the induced metrics on horospheres in X. Such analysis, a generalization of W.Floyd construction of the group completion [Fl], and the described above analysis of geometrical finiteness in X are crucial components of our approach to constructing canonical homeomorphisms of subsets in the sphere at infinity \overline{X} , which induce type-preserving isomorphisms φ: G → H of geometrically finite groups G and H (not necessarily lattices). In particular, as a first result in this direction, we have the following isomorphism theorem [A12]:

Theorem 5.2. Let φ : G → H be a type preserving isomorphism of two nonelementary geometrically finite discrete subgroups 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.

Upon existence of such equivariant homeomorphism f_{φ}, the above problem would be reduced to the question whether f_{φ} is quasisymmetric with respect to the Carnot–Carathéodory metric. In the case of an affirmative answer, it may be possible to find a global G-equivariant homeomorphism (of the sphere at infinity ∂X or even the whole space \overline{X} inducing the isomorphism φ. However, in contrast to the real hyperbolic case, here we have an interesting fenomenon related to possible noncompactness of the boundary ∂M(G) = Ω(G)/G. Namely, even for the simplest case of 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 [Mn]. So it may be possible to (affirmatively) answer the following problem:

Problem 5.3. When are there quasisymmetric homeomorphisms in a Carnot– Carathéodory space ∂X\{∞} compatible with the action of discrete geometrically finite groups G, H ⊂ Isom X but quasisymmetrically non-extendable to the whole space?

We note that, besides the metrical (quasisymmetric) part of this problem, some topological obstructions for extensions of equivariant homeomorphisms f_{φ} of the limit sets, f_{φ}:Λ(G) → Λ(H), may exist. It follows from the following example.

Example 5.4. 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}.

The obstruction in this example is topological and is due to the fact that the quotient manifolds ℍ_{ℂ}^{2}/G and ℍ_{ℂ}^{2}/H are not homeomorphic. Namely, these complex surfaces are disk bundles over the surface S_{g} and have different Toledo invariants: τ(ℍ_{ℂ}^{2}/G) = 2g − 2 and τ(ℍ_{ℂ}^{2}/H) = 0, see [To].

We remark that some of Carnot–Carathéodory spaces are very rigid. In fact, due to P.Pansu [Pa], any quasiconformal map on the boundary ∂X of a quaternionic or octanionic hyperbolic space X (which are symmetric spaces of rank 1) is necessarily an extension of an isometry in X. This shows that any non-trivial homeomorphism f_{φ} : Λ(G) → Λ(H) (non-isometry) is non-extendable to a quasiconformal map of some open subset of ∂X.

In order to attack the above problem, in particular to construct such non-trivial geometric isomorphisms φ in both rigid (like quaternionic and octanionic spaces) and more flexible (like ℍ_{ℂ}^{n}) spaces, one probably may to generalize our blockbuilding method [A1, A5] (whose usefulness in conformal category has been also demonstrated in [A6, A11]). Another aspect of such constructions should be

Problem 5.5. Develop geometric combination theorems generalizing well known Maskit combination theorems for Kleinian groups, see [Mas, A1].

So far, there are the only simplest versions (free products) of such combination theorems, see [FG]. The general case of free amalgamated product is still unknown even in the complex hyperbolic case, where one can however use a week Maskit combination, see [Be].

As the first steps in investigating Problem 5.1, it appears to be promising to study G-equivariant homeomorphisms in Carnot groups Y = ∂X\{∞}, with Carnot– Carathéodory metric ρ, from the following two classes of embeddings f : A ↪ Y, A ⊂ Y.

The first one consists of well known quasisymmetric embeddings f, see [TV]. The second class consists of embeddings f:A ↪ Y, A ⊂ Y, which generalize the

so-called quasi-Möbius embeddings introduced in conformal category by V.Aseev [As, AsT] and J.Väisälä [V]. They have bounded distortion of the cross-ratio (or, equivalently, of the complex cross-ratio in the sense of Koranyi-Reimann [CR4]),

CR(q) = ρ(x_{1}, x_{2})ρ(x_{3}, x_{4})ρ(x_{1}, x_{3})^{−1}ρ(x_{2}, x_{4})^{−1} ,  (5.6)

of quadruples q = {x_{1}, x_{2}, x_{3}, x_{4}} ∈ (Y)^{4} in the following sense.

Let ω:ℝ_{+} → ℝ_{+} be a given homeomorphism, ω(0) = 0. Then, for any quadruple q ∈ A^{4} and its f-image f(q) = {f(x_{1}, …, f(x_{4}} ∈ Y^{4}, we require that

CR(f(q)) ≤ ω(CR(q)) ,  (5.7)

and call such embeddings f quasi-CR embeddings. By continuity, the cross ration can trivially be extended to the one-point compactification Y ∪ {∞}, so we can consider embeddings which are not necessarily fixing ∞.

In particular, for homeomorphism ω(t) = M · η_{α}(t) with given constants M > 0 and α ≥ 1 where

η_{α}(t) = \{ \begin{matrix}t^{α}, for t ≥ 1 \\ t^{1/α}, for 0 ≤ t < 1 , \end{matrix}

we call such embeddings f satisfying (5.7) as (M, α)-CR-embeddings.

In particular, this approach can be used to study the geometric realization Problem 5.1 in the case of free geometrically finite groups G, H ⊂ Isom X whose limit sets are discontinua. Here we have the following

Conjecture 5.8. An embedding f : A ↪ \overline{Y}, A ⊂ \overline{Y}, in a Carnot–Carathéodory space Y is quasi-CR if and only if f preserves the notion of bounded μ-density of subsets Σ ⊂ A.

Here a set Σ is called μ-dense for some μ > 1 if any two points a, b ∈ Σ can be connected by a μ-chain {x_{i}} of points x_{i} ∈ Σ where lim_{i→−∞} x_{i} = a, lim_{i→+∞} x_{i} = b and ln(CR(a, x_{i}, x_{i+1}, b)) ≤ ln μ. Naturally, the μ-density property is a characteristic property of the limit discontinua for geometrically finite groups. We also suspect that that property distinguishes parabolic subgroups of the groups G and H which cannot be quasiconformally conjugate (we expect that they should have non-maximal rank). On the other hand, we expect that an embedding f : A ↪ \overline{Y} of a μ-dense set A in a Carnot–Carathéodory space Y is quasi-CR if and only if it is an (M, α)-CR embedding with some finite M and α. One can expect an affirmative solution of the above problems in the case of geometrically finite groups whose limit discontinua consist only of conical points (for the Heisenberg group Y, compare [Mn]). For study of these problems, one can use methods developed by Koranyi and Reimann [KR1-4], Vodop’yanov [Vo1,VG] as well as a generalization of the technique known as Sullivan’s microscope (see the next section).

6. Deformations of discrete groups and Sullivan’s stability

Here we concern with problems related to deformation spaces of a geometrically finite discrete group G ⊂ P U(n, 1) acting by isometries in the complex hyperbolic space ℍ_{ℂ}^{n} and to the stability theorem of D.Sullivan [Su3] (which has been originally proved for planar Kleinian groups, see also [A11, MaG]). We restrict our attention to the complex hyperbolic case because other rank 1 spaces with negative variable curvature such as quaternionic and octanionic hyperbolic spaces are more rigid. In particular, due to Corlette’s rigidity theorem for harmonic maps [C2], finite volume manifolds locally modeled on these spaces are super-rigid, completely analogous to Margulis super-rigidity in higher rank, see [Mg1]. Moreover, due to Gromov and Schoen [GS], all such finite volume manifolds are arithmetic.

Let G ⊂ P U(n, 1) be a geometrically finite discrete group. Then we have a fundamental problem concerning the deformation space of the inclusion (or of the representation of the group G obtained by restricting a natural inclusion P U(n, 1) → P U(n + 1, 1) to G). The corresponding problem for deformation spaces of a hyperbolically rigid real hyperbolic lattice G ⊂ O(n, 1), n ≥ 3, has its roots in the first construction by the author [A2] of non- trivial curves in the Teichmüller space T (G) of conjugacy classes of representations of G in O(n + 1, 1). Then the situation was greatly clarified by Thurston’s “Mickey Mouse” example [Th] showing that such deformations of a hyperbolic surface are in fact bendings along geodesics. Since that time, several authors studied such deformation spaces, see for example [A4, A6, A8, A9-A11, JM, Ko].

In the case of deformations of a complex hyperbolic group G ⊂ P U(n, 1), we would like to assume that G is not a uniform lattice, that is the quotient ℍ_{ℂ}^{n}/G is noncompact. Otherwise, due to a fundamental result on local rigidity of deformations [GoM], the set of “Fuchsian” representations of G, {Ad h: G → hGh^{−1}, h ∈ P U(n + 1, 1)}, is a connected component R_{0} of the variety of representations Hom(G, P U(n + 1, 1).

On the other hand, we may deform an isomorphic uniform lattice G′ ⊂ P O(2, 1), as well as another isomorphic convex co-compact groups G′ ⊂ P U(2, 1), G′ ≅ G, if such groups have non-elementary subgroups preserving totally geodesic real planes [AG1]. Such quasiconformal deformations bend the corresponding complex surfaces along any simple closed geodesic in its totally real geodesic 2-dim subsurface. For another deformations of groups G′ ⊂ P O(2, 1), see also [GP2]. We can also deform the above groups G′ ⊂ P O(2, 1) in P O(3, 1) ⊂ P U(3, 1), so it makes sense to consider such deformations in higher dimensions, as it has been done by the author in the real hyperbolic case, compare [A9-A11].

The main problem we deal with is as follows.

Conjecture 6.1. Let G ⊂ P U(n, 1) be a geometrically finite group which is either convex cocompact or having parabolic subgroups of rank at least 3. Then every its representation ρ:G → P U(n+k, 1), k ≥ 0, that close enough to a natural inclusion, is in fact discrete and faithful and, furthermore, is a quasiconformal conjugation. That means that there is an equivariant quasiconformal self-homeomorphism f of the extended Heisenberg group \overline{H^{n+k}} such that ρG = f_{∗}G = fGf^{−1}.

This conjecture generalizes a remarkable structural stability theorem of D. Sullivan [Su3] for Kleinian subgroups G in P SL(2, ℂ), which shows that an algebraic structural stability for holomorphic perturbations implies a hyperbolicity property for the action of G on its limit set. In conformal category, we refer to [A11] for our proof of its high dimensional analog. The complex case provides a possibility to generalize a crucial Sullivan’s argument, the so-called λ-Lemma proved by Mañe, Sad and Sullivan [MSS]. Here topological stability is an intermediate open problem. We note also that the negative curvature property is crucial for algebraic stability of convex cocompact subgroups, due to G.Martin [MaG]:

Theorem 6.2. Let {G_{t} : t ∈ ℝ} be a continuous deformation of a torsion free convex cocompact group G ⊂ Isom X in a space X of pinched negative curvature, with convex cocompact G_{t}. Then {G_{t}: t ∈ ℝ}, and its closure in the topology of algebraic convergence in Isom X, consists entirely of isomorphic groups.

For geometrically finite groups without parabolics, it is possible to use a generalization of the so-called Sullivan’s microscope (see [Su2, A11]) to construct probable quasiconformal conjugation of the group actions on their limit sets. It is possible because one can find expanding Heisenberg coverings of the limit set of such a group, which defines a good Cauchy–Riemannian dynamics there. As another result important for a generalization of Sullivan’s arguments, we mention an uniformization theorem for Cauchy–Riemannian structures which is due to Falbel and Gusevskii [FG].

Finally, we would like to mention another problems which are linked with Conjecture 6.1 and, in dimension n = 2, with Problem 3.14 about geometrical finiteness. These problems are related to the boundaries of the deformation spaces R(G, k) = Hom(G, P U(n + k, 1) and T (G, k) = R(G, k)/P U(n + k, 1), k ≥ 0, to the number of their connected components, discreteness and geometrical finiteness of boundary representations. In particular:

Problem 6.3. Is any representation ρ obtained as the limit of geometrically finite representations ρ_{i} ∈ R(G, k) discrete and faithful, that is ρ(G) ⊂ P U(n + k) is a discrete group isomorphic to a given geometrically finite group G ⊂ P U(n, 1)?

We note that, due to G.Martin [MaG], the subset of discrete faithful representations is closed in R(G, k).

Another question is about the boundary of the Teichmüller space T (G, k) of a convex cocompact group G ⊂ P U(n, 1), n ≥ 2, k ≥ 0.

Problem 6.4. Are there convex cocompact faithful representations ρ_{t} ∈ R(G, k), t ∈ ℝ, of a convex cocompact group G ⊂ P U(n, 1) which converge to a boundary discrete representation ρ whose image ρ(G) ⊂ P U(n + k) has accidental parabolic elements?

We remark that I.Belegradek [Be] constructed a discrete faithful representation ρ of the fundamental group G of a compact hyperbolic surface into P U(2, 1) such that ρ(G) has parabolic elements. This construction uses a Maskit combination theorem, and it is unclear whether ρ(G) lies in the boundary ∂R_{0}(G) of a component of the variety of convex co-compact representations G → P U(2, 1). However the Teichmüller space T (G) = T (G, 0) does have “cusps” due to a recent result by Apanasov and Gusevskii [AG2]:

Theorem 6.5. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be a uniform lattice isomorphic to the fundamental group of a closed surface S_{g} of genus p ≥ 2. Then, for any simple closed geodesic α ⊂ S_{p} = H_{ℝ}^{2}/G, there is a continuous deformation ρ_{t} = f_{t}^{∗} induced by G-equivariant quasiconformal homeomorphisms f_{t} : ℍ_{ℂ}^{2} → ℍ_{ℂ}^{2} whose limit representation ρ_{∞} corresponds to a boundary cusp point of the Teichmüller space T (G). In other words, the boundary group ρ_{∞}(G) has an accidental parabolic element ρ_{∞}(g_{α}) where g_{α} ∈ G represents the simple closed geodesic α ⊂ S_{p}.

We note that, due to our construction of such continuous quasiconformal deformations in [AG2], such deformations are independent if the corresponding simple closed 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 6.6. Let G ⊂ P O(2, 1) ⊂ P U(2, 1) be as in Theorem 6.5. Then there is a continuous deformation R : ℝ^{2p−2} → T (G) whose boundary group G_{∞} = R(∞)(G) has 2p − 2 non-conjugate accidental parabolic subgroups.

Finally, we mention another aspect of the intrigue Problem 3.14:

Problem 6.7. Construct a geometrically infinite (finitely generated) group G ⊂ P U(n, 1), n ≥ 2, whose limit set is the whole sphere at infinity, Λ(G) = ∂ℍ_{ℂ}^{n} = \overline{H^{n}}, and which is the limit of convex cocompact groups G_{i} ⊂ P U(n, 1) from the Teichmüller space T (G) of a convex cocompact group G ⊂ P U(n, 1). Is that possible for a Schottky group G?

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