Stability of Kleinian groups

Katsuhiko Matsuzaki ^{∗} Department of Mathematics, Ochanomizu University Tokyo, Japan

We survey stability of Kleinian groups. Several results in this note are also contained in the forthcoming monograph [7].

Let Γ be a finitely generated non-elementary Kleinian group, which is identified with a discrete subgroup of PSL_{2}(C), with a fixed system of generators Γ = ⟨γ_{1}, …, γ_{N}⟩. We consider the set of PSL_{2}(C)-representations

Hom(Γ) = {ρ | ρ : Γ → PSL_{2}(C) is a homomorphism}.

This is regarded as an analytic subset of PSL_{2}(C)^{N} by the correspondence

ρ ↦ (ρ(γ_{1}), …, ρ(γ_{N})) ∈ PSL_{2}(C)^{N}.

We say that Γ is structurally stable if there is a neighborhood U of the identity representation id in Hom(Γ) such that any ρ ∈ U is a faithful representation.

However, a weaker condition than structural stability is more interesting in deformation theories of Kleinian groups, where we treat an analytic set of all representations sending any parabolic element to a parabolic one or the identity. Letting

PHom(Γ) = {ρ ∈ Hom(Γ) | tr^{2}ρ(γ) = 4 for any parabolic γ ∈ Γ},

we call Γ weakly structurally stable if the condition of structural stability is satisfied after replacing Hom(Γ) with PHom(Γ).

Here, the property that any ρ ∈ U ⊂ PHom(Γ) is faithful is actually equivalent to that ρ is a quasiconformal deformation. Indeed, we can apply the λ-lemma to a holomorphic family of isomorphisms defined over a complex disk holomorphically embedded in U that passes a given point of U. Thus the weakly structural stability is nothing but the following quasiconformal stability:

∗E-mail: matsuzak@math.ocha.ac.jp

Let T (Ω(Γ)/Γ) be the Teichmüller space of the union of orbifolds Ω(Γ)/Γ. For every [μ] ∈ T (Ω(Γ)/Γ), we denote by f_{μ} a quasiconformal automorphism of Ĉ that gives the deformation [μ] of the complex structure of Ω(Γ)/Γ and that satisfies a suitable normalization. Then a holomorphic map

Ψ̃ : T (Ω(Γ)/Γ) × PSL_{2}(C) → PHom(Γ)

is defined by the conjugation of A ∘ f_{μ}, for any pair ([μ], A) ∈ T (Ω(Γ)/Γ) × PSL_{2}(C). It is known that Ψ̃ is well-defined. Moreover, the Sullivan rigidity theorem implies that the image of Ψ̃ coincides with the set of the whole representations induced by quasiconformal automorphisms of Ĉ (cf. [7, Chapter 5]). This set is called the quasiconformal deformation space and denoted by QHom(Γ) (⊂ Hom(Γ)). We say that Γ is quasiconformally stable if there is a neighborhood U of the identity representation id such that

PHom(Γ) ∩ U ⊂ QHom(Γ).

Gardiner and Kra [3] investigated the derivative dΨ̃|_{id} of Ψ̃ at id, representing the Zariski tangent space of the analytic set PHom(Γ) as Eichler cohomology. This is called the Bers map. They showed that dΨ̃|_{id} is injective and decomposed the tangent space into subspaces caused by PSL_{2}(C)- conjugations, quasiconformal deformations and deformations of projective structure. In particular, they proved that surjectivity of dΨ̃|_{id} implies quasiconformal stability.

For a torsion-free Kleinian group, Marden [5] proved that if Γ is geometrically finite then it is quasiconformally stable, by investigation of fundamental polyhedra. Later Sullivan [10] proved the converse, namely, that quasiconformal stability implies geometric finiteness, by 3-dimensional topological arguments to compare the dimension of the Teichmüller space T (Ω(Γ)/Γ) with the dimension of PHom(Γ).

In [6], we extend this equivalence to Kleinian groups with torsion. One direction is easy if we pass Γ to a torsion-free subgroup of finite index by the Selberg lemma, but the other not. We consider a core in the 3-dimensional hyperbolic orbifold H^{3}/Γ, where a core means a compact suborbifold such that the inclusion induces an isomorphism between the orbifold fundamental groups. Moreover, we require that the core is relative to the boundary at infinity Ω(Γ)/Γ. If we can construct such a core, then we can obtain a relation between dim T (Ω(Γ)/Γ) and dim PHom(Γ) at id because both are topological quantities determined only by the topology of the relative core.

We can prove the existence of an orbifold relative core in the case that Γ is indecomposable as a free product in a certain sense. For the general case, we decompose Γ into indecomposable ones and use induction arguments to compare the dimensions.

Theorem 1 The following conditions are equivalent for any finitely generated non-elementary Kleinian group Γ:

1. Γ is geometrically finite;

2. Γ is quasiconformally stable;

3. the Bers map dΨ̃|_{id} is an isomorphism.

In particular:

Corollary 2 If Γ is geometrically finite, then QHom(Γ) is a complex regular submanifold of PSL_{2}(C)^{N}.

Next we will prove that Corollary 2 is actually satisfied for any finitely generated Kleinian group. Since Ψ̃ is a holomorphic immersion onto QHom(Γ), the only problem is compatibility of the Teichmüller topology of T (Ω(Γ)/Γ)× PSL_{2}(C) and the topology of QHom(Γ), which is the algebraic topology for PSL_{2}(C)-representations.

Problem If ρ_{n} converge to id in QHom(Γ), do there always exist t_{n} ∈ Ψ̃^{−1}(ρ_{n}) such that t_{n} converge to the base point 0 ∈ T (Ω(Γ)/Γ) as n → ∞ ?

This problem is originated in Bers [1, p.578], where it was announced that the proof would appear elsewhere, however it has not appeared as far as the author knows. See also [2]. Later Krushkal published a series of papers (cf. [4]) concerning this problem. A finitely generated Kleinian group is called conditionally stable or quasi-stable if it satisfies the property in the problem above.

We will show that a result on geometric convergence of Kleinian groups yields the affirmative answer to this problem.

Theorem 3 Any finitely generated Kleinian group Γ is conditionally stable. Hence QHom(Γ) is a complex regular submanifold of PSL_{2}(C)^{N}.

We first remark the following two facts.

Lemma 4 Γ is conditionally stable if any component subgroup of Γ is conditionally stable.

Proof. This follows easily from the definition of conditional stability.

Lemma 5 Let Γ be a finitely generated Kleinian group and Γ′ a subgroup of Γ of finite index. If Γ′ is conditionally stable, then so is Γ.

Proof. Suppose that Γ is not conditionally stable. Then there is a sequence ρ_{n} ∈ QHom(Γ) converging to id such that the maximal dilatation of the extremal quasiconformal automorphism f_{n} inducing ρ_{n} does not tend to 1 as n → ∞. Here the extremal quasiconformal map is the one with the smallest maximal dilatation among quasiconformal maps with the required property.

We restrict ρ_{n} to the subgroup Γ′ and have ρ′_{n} ∈ QHom(Γ′). Then ρ′_{n} converges to id and f_{n} induces ρ′_{n}. Since Γ′ is of finite index in Γ, f_{n} is also the extremal quasiconformal automorphism that induces ρ′_{n} (cf. Ohtake [9]). But this contradicts the assumption that Γ′ is conditionally stable. Thus we see that Γ is also conditionally stable.

By these facts, it suffice to consider torsion-free function groups Γ for proving Theorem 3. It is known that such Γ is constructed from elementary groups, quasifuchsian groups and totally degenerate groups without APT by a finite number of applications of the Maskit combination theorem. Moreover we can see that conditional stability is preserved under the Maskit combination theorem:

Lemma 6 Assume that a torsion-free function group Γ is constructed from Γ_{1} and Γ_{2} (as the amalgamated free product or the HNN–extension) by the Maskit combination theorem. If both Γ_{1} and Γ_{2} are conditionally stable, then so is Γ.

Proof. See [7, Section 7.3].

Therefore Theorem 3 will complete if it is solved for totally degenerate groups without torsion nor APT. The crucial fact for this step is the following result due to Thurston (cf. Ohshika [9] and [7, Section 7.2]).

Proposition 7 Let Γ_{0} be a finitely generated torsion-free Fuchsian group and θ_{n} : Γ_{0} → Γ_{n} a sequence of type-preserving isomorphisms onto Kleinian groups, which converges algebraically to a type-preserving isomorphism θ : Γ_{0} → Γ. If Γ is a totally degenerate group, then Γ_{n} also converge geometrically to Γ.

Applying this proposition, we can assert:

Lemma 8 Under the same circumstances as in Proposition 7, if Γ_{n} and Γ are totally degenerate groups, then the marked complex structures t_{n} of Ω(Γ_{n})/Γ_{n} converge to t of Ω(Γ)/Γ. In particular, any torsion-free, totally degenerate group without APT is conditionally stable.

Proof. Let C_{n} be the convex core of the hyperbolic manifold H^{3}/Γ_{n} and ∂C_{n} the relative boundary of C_{n}, which is regarded as a pleated surface with a marked hyperbolic structure s_{n}. By Proposition 7, we can see that s_{n} converge to the marked hyperbolic structure s of the boundary surface of the convex core of H^{3}/Γ. By Sullivan’s theorem (cf. [7, Section 7.1]), s_{n} and t_{n} are in a bounded Teichmüller distance independent of n. Hence {t_{n}} is a bounded sequence in the Teichmüller space and there is a subsequence {t_{n′}} which converges to some t′. Then Γ_{n′} converge algebraically to a bgroup Γ′ such that the marked complex structure of D′/Γ′ is t′, where D′ is the invariant component of Ω(Γ′). However, Γ′ should coincide with Γ, and hence t′ = t.

Thus we obtain Theorem 3.

References

[1] Bers, L. On boundaries of Teichmüller spaces and on kleinian groups I. Ann. of Math., 91 (1970), 570–600.

[2] Bers, L. Spaces of Kleinian groups. Maryland conference in several complex variables. Lecture Notes in Math. 155, Springer, pp. 9–34.

[3] Gardiner, F. and Kra, I. On stability of Kleinian groups. Indiana Univ. Math. J., 21 (1972), 1037–1059.

[4] Krushkal, S. Quasiconformal stability of Kleinian groups. Siberian Math. J., 20 (1979), 229–234.

[5] Marden, A. The geometry of finitely generated Kleinian groups. Ann. of Math., 99 (1974), 383–462.

[6] Matsuzaki, K. Structural stability of Kleinian groups. Michigan Math. J., 44 (1997). To appear.

[7] Matsuzaki, K. and Taniguchi, M. The theory of Kleinian groups. Oxford Univ. Press. To appear.

[8] Ohshika, K. Divergent sequences of Kleinian groups. Preprint.

[9] Ohtake, H. Lifts of extremal quasiconformal mappings of arbitrary Riemann surfaces. J. Math. Kyoto Univ., 22 (1982), 191–200.

[10] Sullivan, D. Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups. Acta Math., 155 (1985), 243–260.