数理解析研究所講究録1022 巻 1997 年 87-92

87

Stability of Kleinian groups
Katsuhiko Matsuzaki
松崎*克彦

Department of Mathematics, Ochanomizu UniversityTokyo, JapanWe survey stability of Kleinian groups. Several results in this note arealso contained in the forthcoming monograph [7].
Let be a finitely generated non-elementary Kleinian group, which is, with a fixed system of genidentified with a discrete subgroup of(C)-representations. We consider the set oferators$\Gamma$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$\Gamma=\langle\gamma_{1}, \ldots , \gamma_{N}\rangle$$\mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)=${$\mathrm{P}\mathrm{S}\mathrm{L}_{2}$$\rho|\rho$:$\Gammaarrow \mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$This is regarded as an analytic subset ofis a homomorphism}.

$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{c})N$by the correspondence$\rho\mapsto(\rho(\gamma_{1}), \ldots, \beta(\gamma N))\in \mathrm{P}\mathrm{s}\mathrm{L}_{2}(\mathrm{C})^{N}$

We say that is structurally stable if there is a neighborhood of the identity
representation $id$ i.n
is a faithful representation.
such that any
However, a weaker condition than structural stability is more interestingin deformation theories of Kleinian groups, where we treat an analytic set of
all representations sending any parabolic element to a parabolic one or theidentity. Letting$U$$\Gamma$$\rho\in U$$\mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)$

PHomwe call$(\Gamma)=${$\rho\in \mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)|\mathrm{t}\mathrm{r}^{2}\rho(\gamma)=4$

for any parabolic$\gamma\in\Gamma$},weakly structurally stable if the condition of structural stabil.ity iswith PHom .
satisfied.after replacing
is faithful is actually
Here, the property that any
equivalent to that is a quasiconformal deformation. Indeed, we can apply$\Gamma$$(\Gamma)$$\mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)$$\rho\in U\subset \mathrm{P}\mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)$$\rho$-mail: matsuzak@math.ocha.ac.jp$*\mathrm{E}-$88the -lemma to a holomorphic family of isomorphisms defined over a complex
disk holomorphically embedded in that passes a given point of . Thus
the weakly structural stability is nothing but the following quasiconformal
stability:.
be the Teichm\"uller space of the union of orbifolds
Let, we denote by a quasiconformal automorphism
For every
andof the complex structure ofof that gives the deformationthat satisfies a suitable normalization. Then a holomorphic map$\lambda$

$U$

$U$

$\Omega(\Gamma)/\Gamma$$T(\Omega(\mathrm{r})/\Gamma)$$[\mu]\in T(\Omega(\mathrm{r})/\Gamma)$$f_{\mu}$$\hat{\mathrm{C}}$$\Omega(\Gamma)/\Gamma$$[\mu]$$\tilde{\Psi}$:$T(\Omega(\mathrm{r})/\Gamma)\cross \mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})arrow \mathrm{P}\mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{r})$, for any pairis defined by the conjugation of. It is known that is well-defined. Moreover, the Sullivan rigidity
theorem implies that the image of coincides with the set of the whole representations induced by quasiconformal automorphisms of (cf. [7, Chapter5]). This set is called the quasiconformal deformation space and denoted by. We say that is quasiconformally stable if there isQHom(F)a neighborhood of the identity representation $id$ such that$([\mu], A)\in T(\Omega(\mathrm{r})/\Gamma)\cross$$A\circ f_{\mu}$$\tilde{\Psi}$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$\tilde{\Psi}$$\hat{\mathrm{C}}$$\Gamma$$(\subset \mathrm{H}\mathrm{o}\mathrm{m}(\mathrm{r}))$$U$

PHom$(\Gamma)\cap U\subset \mathrm{Q}\mathrm{H}\mathrm{o}\mathrm{m}(\Gamma)$.

of at $id$ , repGardiner and Kra [3] investigated the derivativeas Eichler
resenting the Zariski tangent space of the analytic set PHom
is injeccohomology. This is called the Bers map. They showed thattive and decomposed the tangent space into subspaces caused byconjugations, quasiconformal deformations and deformations of projective
implies quasiparticular, they proved that surjectivity ofstructure.
conformal stability.
For a torsion-free Kleinian group, Marden [5] proved that if 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 topologicalarguments to compare the dimension of the Teichm\"uller
with the dimension of PHom(F).
In [6], we extend this equivalence to Kleinian groups with torsion. Onedirection is easy if we pass to a torsion-free subgroup of finite index by theSelberg lemma, but the other not. We consider a core in the 3-dimensional, where a core means a compact suborbifold such
hyperbolic orbifold$\tilde{\Psi}$$d\tilde{\Psi}|_{id}$$(\Gamma)$$d\tilde{\Psi}|_{id}$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})-$$\ln$$d\tilde{\Psi}|_{id}$$\Gamma$.ace$\mathrm{s}\mathrm{p}.$$\Gamma$$\mathrm{H}^{3}/\Gamma$$T(\Omega(\Gamma)/\Gamma)$

89that the inclusion induces an isomorphism between the orbifold fundamentalgroups. Moreover, we require that the core is relative to the boundary at. If we can construct such a core, then we can obtain a relation
infinity
betweenand $\dim$ PHom(F) at $id$ because both are topologicalquantities determined only.by the topology of the relative core.
We can prove the existence of an orbifold relative core in the case thatis indecom.posable as a free product in a certain sense. For the general case,
we decompose into indecomposable ones and use induction arguments tocompare the dimensions.
$\Omega(\Gamma)/\Gamma$$\dim T(\Omega(\mathrm{r})/\Gamma)$$\Gamma$$\Gamma$

Theorem 1 The following conditions are equivalent for any finitely generated $non- e\iota ementaw$ Kleinian group :1. is geometrically $finite_{j}$.
$\Gamma$$\Gamma$2.

$\Gamma$is quasiconformally stablej

3. the Bers map$d\tilde{\Psi}|_{id}$is an isomorphism.

In particular:

Corollary 2 If

submanifold of$\Gamma$

is geometrically $finite_{f}$ then QHom(F) is a complex regular$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})^{N}$

Next we will prove that Corollary 2 is actually satisfied for any finitely
generated Kleinian group. Since is a holomorphic immersion onto QHom ,the only problem is compatibility of the Teichm\"uller topology ofand the topology of QHom , which is the algebraic topology for$\tilde{\Psi}$$(\Gamma)$$T(\Omega(\mathrm{r})/\Gamma)\cross$$(\Gamma)$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}$(C)-representations.

Problem lf
converge to $id$ in QHom(F), do there always exist
such that converge to the base pointas$\rho_{n}$$\tilde{\Psi}^{-1}(\rho_{n})$$\mathrm{O}\in T(\Omega(\Gamma)/\Gamma)$$t_{n}$$t_{n}\in$$narrow\infty$?

This problem is originated in Bers [1, p.578], where it was announced thatthe proof would appear elsewhere, however it has not appeared as far as theauthor knows. See also [2]. Later Krushkal published a series of papers (cf.
[4] concerning this problem. A finitely generated Kleinian group is called$Conditional\iota_{y}$ 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.
$)$.,

90Theorem 3 Any finitely generated Kleinian group is conditionally stable.
Hence QHom(F) is a complex regular submanifold of$\Gamma$$\mathrm{P}\mathrm{S}\mathrm{L}_{2}(\mathrm{C})^{N}$

We first remark the following two facts.

Lemma 4 is conditionally stable if any component subgroup ofditionally stable.

$\Gamma$$\Gamma$

is con-Proof. This follows easily from the definition of conditional stability.
Lemma 5 Let be a finitely generated Kleinian group andof finite index. If is conditionally stable, then so is .
$\Gamma$a subgroup of$\Gamma$$\Gamma’$$\Gamma$$\Gamma’$$\blacksquare$is not conditionally stable. Then there is a sequence
QHom(F) converging to $id$ such that the maximal dilatation of theextremal quasiconformal automorphism inducing does not tend to 1 as$narrow\infty$ . Here the extremal quasiconformal map is the one with the smallest
maximal dilatation among quasiconformal maps with the required property.
QHom(F’). Thenand haveto the subgroupWe restrictis alsoconverges to $id$ and induces . Since is of finite index in(cf. Ohtake [9]).
the extremal quasiconformal automorphism that induces
But this contradicts the assumption that is conditionally stable. Thus wesee that is also conditionally stable.
By these facts, it suffice to consider torsion-free function groups forproving Theorem 3. It is known that such is constructed from elementarygroups, 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:

Proof. Suppose that$\Gamma$$\rho_{n}\in$$f_{n}$$\rho_{n}$$\Gamma’$$\rho_{n}’$$\rho_{n}’\in$$\rho_{n}$$f_{n}$$\Gamma’$$\rho_{n}’$$\Gamma,$$f_{n}$$\rho_{n}’$$\Gamma’$$\blacksquare$$\Gamma$$\Gamma$$\Gamma$Lemma 6 Assume that a torsion-free function group is constructed from(as the amalgamated free product or the $HNN$-extension) by theandare conditionally stable, thenand
Maskit combination theorem. If bothso is F.
$\Gamma$$\Gamma_{1}$$\Gamma_{2}$$\Gamma_{1}$Proof. See [7, Section 7.3].

$\Gamma_{2}$$\blacksquare$

Therefore Theorem 3 will complete if it is solved for totally degenerategroups 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]).

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Proposition 7 Let
be a finitdy generated torsion-free Fuchsian groupa sequence of type-preserving isomorphisms onto Kleinian:andgroups, which converges algebraically to a type-preserving isomorphism :also converge geometri. If is a totally degenerate group, thencally to .
$\Gamma_{0}$$\Gamma_{0}arrow\Gamma_{n}$$\theta_{n}$$\theta$$\Gamma_{n}$$\Gamma$$\Gamma_{0}arrow\Gamma$$\Gamma$

Applying this proposition, we can assert:
Lemma 8 Under the same circumstances as in Proposition
if andare totally degenerate $group_{S}$ , then the marked complex structuresof. In particularf any torsion-free, totallyconverge to ofdegenerate group without APT is conditionally stable.
$7_{f}$$\Gamma_{n}$$t_{n}$$\Gamma$$t$$\Omega(\Gamma_{n})/\Gamma_{n}$$\Omega(\Gamma)/\Gamma$

andbe the convex core of the hyperbolic manifold, which is regarded as a pleated surface withthe relative boundary ofa marked hyperbolic structure . By Proposition 7, we can see thatconverge to the marked hyperbolic structure of the boundary surface of. By Sullivan’s theorem (cf. [7, Section 7.1]),
the convex core ofand are in a bounded Teichm\"uller distance independent of . Henceis a bounded sequence in the Teichm\"uller space and there is a subsequence
converge algebraically to a bwhich converges to some . Thenis , where $D’$ isgroup such that the marked complex structure of. However,should coincide with , andthe invariant component ofhence $t’=t$ .

Proof. Let$\partial C_{n}$$\mathrm{H}^{3}/\Gamma_{n}$$C_{n}$$C_{n}$$s_{n}$$s_{n}$$s$$s_{n}$$\mathrm{H}^{3}/\Gamma$$\{t_{n}\}$$n$$t_{n}$$t’$$\{t_{n’}\}$$\Gamma_{n’}$$D’/\Gamma’$$\Gamma’$$\Omega(\Gamma’)$$\Gamma’$$t’$$\Gamma$$\blacksquare$Thus we obtain Theorem 3.

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References[1] Bers, L. On boundaries of Teichm\"uller
Ann. of Math., 91 (1970), 570-600.

sp.aces and on kleinian groups I.

[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. ofMath., 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.