数理解析研究所講究録 1270 巻 2002 年 93-100

93

DENSITY OF GEOMETRICALLY FINITE KLEINIANGROUPS
KEN’ICHI OHSHIKA

阪 大理

大鹿

健

一

1. INTRODUCTION
In 1988, Thurston published alist of unsolved problems on Kleiniangroups and hyperbolic 3-manifolds in [19]. One of the problems there asks
if every finitely generated Kleinian group is an algebraic limit of geometrically finite ones. This problem can be regarded as ageneralization ofBers’ density conjecture asserting that every -group would be alimit ofquasi-Fuchsian groups with the same conformal structure on invariant components of the regions of discontinuity. Recently, Bromberg [5], using abrilliant idea involving adeformation theory of geometrically finite cone
manifolds, solved most part of Bers’ density conjecture. To be more precise,he proved that every totally degenerate -group without parabolic elements
such that the corresponding hyperbolic 3-rnanifold has asimply-degenerate
end whose neighbourhood has unbounded injectivity radii is an algebraic
limit of quasi-Fuchsian groups. The case of bounded injectivity radii had
been already solved by Minsky [9], [11]. In the present paper, we shall prove
Thurston’s conjecture above for freely indecomposable Kleinian groups thatare not isomorphic to surface groups, using the result of Bromberg, but not
his technique.
$\mathrm{b}$$\mathrm{b}$Theorem 1. Let
be a finitely generated freely indecomposable Kleiniangroup without parabolic elements that is not isomorphic to a surface group.
Then, there are a geometrically finite Kleinian groupsuch thatis homeomorphic toand its quasi-conformal deformations
withisomorphisms:
which converge to an isomorphism :asrepresentations into.
$G$$\Gamma$$\mathrm{H}^{3}/\Gamma$$\mathrm{H}^{3}/G$$\phi_{i}$$\Gammaarrow\Gamma_{i}$$\Gamma_{:}$$\psi$$\Gammaarrow G$$PSL_{2}\mathrm{C}$We have been informed that Brock and Bromberg [4] also proved thistheorem using asimilar argument previous to our work. They also dealt
with the case of doubly degenerate group. Thus, the only remaining caseof Thurston’s conjecture for freely indecomposable Kleinian groups is thatof groups with parabolic elements. We are also informed that Brock andBromberg are working on such Kleinian groups.
We should also note that our main theorem follows from the main theoremof [15] if the ending lamination conjecture is solved affirmatively. Minsky
has been working on the conjecture (see [9], [10], [11]), and we expect that
he will achieve the goal in quite near future

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KEN’ICHI OHSHIKA

2. PRELIMINARIESIn this paper, all Kleinian groups axe assumed to be finitely generated

and torsion ffee. For aKleinian group , we consider the corresponding. The convex submanifold ofcomplete hyperbolic 3-manifoldthat is minimal among the convex deformation retracts is called the convex. AKleinian group is said to, and is denoted bycore ofhas finite volume.
be geometrically finite when the convex core
By Scott’s theorem [17], for aKleinian group , there is acodimension-O
issuch that the inclusion from tocompact submanifold inahomotopy equivalence. The boundary components of correspond bijecsince for each boundary component of , there
tively to the ends ofis aunique end contained in the component of the complement attachedto . We say then that the end faces the boundary component , and also
is said to be geometrically finite whenthat faces . An end of
there is aneighbourhood of that contains no closed geodesies. If an end
facing aboundary component of acompact core is geometrically finite,. The Kleinian groupthen has aneighbourhood homeomorphic to
are geometricallyis geometrically finite if and only if all the ends offinite.
AKleinian group is said to be freely indecomposable when there is nonon-trivial free product decomposition of . Bonahon showed in [2] thatis either geometriis freely indecomposable, every end ofwhencally finite or simply degenerate in the following sense. An end facing
an incompressible boundary component of acompact core is said to beon
simply degenerate when there is asequence of simple closed curves
tends to as
inhomotopic tosuch that the closed geodesic$iarrow\infty$ .
It was also proved that if facing is simply degenerate, then
. Regarding simple closed
has aneighbourhood homeomorphic to,consider their limit, after
wecanas projective laminations on
curvestaking asubsequence, in the projective lamination space, which is compact.
Such alimit projective lamination or ameasured lamination in that class is
said to represent the ending lamination of . If two projective laminations
represent the ending lamination of the same end, then their supports mustcoincide. Moreover, ameasured lamination representing an ending lamination is known to be maximal and connected. Here we say that ameasured
lamination is maximal when it is not aproper sub-lamination of anothermeasured lamination.
For aKleinian group , we denote by $AH(G)$ the set of faithful discrete
modulo conjugacy. We endow $AH(G)$intorepresentations ofwith the quotient topology induced ffom the space of representations withthe topology of point-wise convergence. We represent an element of $AH(G)$where is arepresentation representing the element, and is
as apairin $AH(G)$, which is aKleinian group. An element
its image in$G$$\mathrm{H}^{3}/G$$\mathrm{H}^{3}/G$$G$$C(\mathrm{H}^{3}/G)$$\mathrm{H}^{3}/G$$C(\mathrm{H}^{3}/G)$

$G$

$C$

$C$

$\mathrm{H}^{3}/G$$\mathrm{H}^{3}/G$

$C$

$C$

$S$

$\mathrm{H}^{3}/G$

$e$

$S$

$S$

$e$

$S$

$e$

$e$

$\mathrm{H}^{3}/G$

$e$

$S$

$e$

$G$

$S\cross \mathrm{R}$$e$$\mathrm{H}^{3}/G$

$G$

$G$

$\mathrm{H}^{3}/G$

$G$

$e$

$S$

$\{\gamma_{\dot{1}}\}$$S$$\mathrm{H}^{3}/G$$\gamma_{\dot{1}}^{*}$$\gamma_{\dot{|}}$

$e$

$S$

$e$

$S\cross \mathrm{R}$

$e$

$S$

$[\gamma_{\dot{1}}]$

$e$

$G$

$G$

$PSL_{2}\mathrm{C}$$\Gamma$$(\Gamma, \phi)$$PSL_{2}\mathrm{C}$$\phi$$(\Gamma, \phi)$

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DENSITY OF GEOMETRICALLY FINITE KLEINIAN GROUPS

is said to be aquasi-conformal deformation ofwhen there is aquasiconformal homeomorphism :such that $\phi(g)=fgf^{-1}$ fo all $g\in$. The subspace of $AH(G)$ consisting of all quasi-conformal deformationsofis denoted by $QH(G)$ . It is known, by work of Ahlfors, Bers, Kra,
Maskit and Sullivan, that if
is freely indecomposable, then $QH(G)$ is
homeomorphic to the Teichmiiller space of, wheredenotes theregion of discontinuity of on. This correspondence is given by taking$\in QH(G)$ to the conformal structure on
induced from thenatural conformal structure on
using ahomeomorphism fromtogiven by aquasi-conformal homeomorphism as above. We denotethe inverse of this correspondence by $qc$ : $\mathcal{T}(\Omega c/G)arrow QH(G)$ and call it
the Ahlfors-Bers map.
Let be aclosed surface of genus at least 2. Since admits acomplete
hyperbolic metric, there is afaithful discrete representation of
into. The image of such arepresentation is called aFuchsian group, andits quasi-conformal deformations, regarded as representations to,are called quasi-Fuchsian groups. AKleinian groupis quasi-Fuchsian
if and only if
consists of two components both of which are invariant
by . For aFuchsian group $H$ , as was explained in the last paragraph,the space of quasi-Fuchsian groups modulo conjugacy is homeomorphic to, wheredenotes the Teichmiiller space ofthe marked conformal structures with orientation reversing markings. The
Kleinian group which is the image of afaithful discrete representation ofinto
is said to be a group whenhas only one invariant
component. Furthermore, if
is connected, then is said to be atotally
degenerate -group. We should note that any -group without parabolic
elements must be totally degenerate. Similarly aKleinian group as above
is said to be doubly degenerate if
is empty. If such without parabolic
elements is neither quasi-Fuchsian nor atotally degenerate -group, then it
must be doubly degenerate.
$G$

$f$

$S_{\infty}^{2}arrow S_{\infty}^{2}$$\mathrm{r}$

$G$

$G$

$G$

$\Omega c/G$

$G$$\Omega c$$S_{\infty}^{2}$$(\Gamma, \phi)$$\Omega_{G}/G$$\Omega_{G}/G$$\mathrm{f}\mathrm{i}\mathrm{r}/\mathrm{r}$$\Omega_{\Gamma}/\Gamma$

$f$

$S$

$S$

$\pi_{1}(S)$$PSL_{2}\mathrm{R}$$PSL_{2}\mathrm{C}$$G$$\Omega_{G}$$G$$\mathcal{T}(\Omega_{H}/H)\cong \mathcal{T}(S)\cross \mathcal{T}(\overline{S})$$\mathcal{T}(\overline{S})$$G$$\pi_{1}(S)$$PSL_{2}\mathrm{C}$$\mathrm{b}$$\Omega_{G}$$G$$\Omega_{G}$$\mathrm{b}$$\mathrm{b}$

$G$

$G$

$\Omega_{G}$$\mathrm{b}$3. PrOOf OF THE MAIN THEOREM

be acompact core of. By Bonahon’s theorem, each endofis either geometrically finite or simply degenerate. In particular,
is homeomorphic to the interior of . By Thurston’s uniformization
theorem for atoroidal compact 3-manifolds with boundary (see Thurston[20] and Morgan [12] , there is ageometrically finite Kleinian group without
parabolic elements such thatis homeomorphic to. Let :
be an isomorphism induced by ahomeomorphism as above, andregard $(G, \psi)$ as an element of $AH(\Gamma)$ . We are to define asequence ofquasi-conformal deformations, which will be proved to converge$(G,\psi)$
algebraically toeventually.
,Let ,be the boundary components of , which are incompressible since is freely indecomposable. We number them so that ,
,
Let

$C$

$\mathrm{H}^{3}/G$$\mathrm{H}^{3}/G$$C$$\mathrm{H}^{3}/G$$)$$\Gamma$$\mathrm{H}^{3}/G$$\mathrm{H}^{3}/\Gamma$$\psi$$\Gammaarrow G$$\{(\Gamma:, \phi_{i})\}$$S_{1}$$\ldots$$G$$S_{k}$$C$$S_{1}$$\ldots$$S_{\kappa}$

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KEN’ICHI OHSHIKA

among them face simply-degenerate ends, whereas the rest face geomet, for otherwise
itself isrically finite ends. We can assume that, and letbgeometrically finite. Consider acomponent $Sj$ withfacing $Sj$ .
the end ofof
corresponding to the subgroupof
Take asubgroup$G$$\kappa\geq 1$$e^{S_{j}}$$j\leq\kappa$$\mathrm{e}$$\mathrm{H}^{3}/G$$G^{S_{j}}$$\pi_{1}(C)\cong G$$G$$\pi_{1}(S_{j})$.

Lemma 2. For jgroup.

$\leq\kappa$, the Kleinian groupis a totally degenerate b-$G^{S_{j}}$no parhas no parabolic elements,
is either quasi-Fuchsian or atotallyabolic elements either. Therefore,
degenerate -group or adoubly degenerate group. Since there is aneigh,that can be lifted homeomorphically tobourhood of the end
simply, the manifold
must have at least one
which we denote byquasi-Fuchsian.
beend;cannothence
degenerateis doubly degenerate.
Suppose now, seeking acontradiction, that
. Let:
other thanThen
has asimply-degenerate end
theinclusion.
toprojection
associatedcovering
be theBy the covering theorem due to Thurston (see also Canary [6]), there is a
is proper and acovering map toneighbourhood $E’$ of such thatover . Also, it is impossible
bundleis not asurface
its image since
since the ending laminations for andthat $E’$ covers aneighbourhood of
has asimply-degenerate end distinct
fordiffer. It follows that
, whose neighbourhood is covered by $E’$ . Let $S’$ be the boundary
ffom
component of $C$ facing . Then has aneighbourhood homeomorphic totending to , which
J. It follows that pleated surfaces :tending to ,can be obtained by projecting pleated surfaces in$S’$ .
Therefore, $S’$ must be
are homotopic to afinite-sheeted covering by
homotopic to afinite-sheeted covering of . This is impossible, as can beseen by elementary 3-dimensional topology, since we assumed that $C$ is not.
homeomorphic to

Proof. Since we assumed that

$G^{S_{j}}\mathrm{h}\mathrm{s}$$G$$G^{S_{j}}$$e^{S_{j}}$$\mathrm{H}^{3}/G^{S_{j}}$$\tilde{e}^{S_{j}}$$\mathrm{H}^{3}/G^{S_{\mathrm{j}}}$$G^{S_{j}}$$G^{S_{\dot{f}}}$$\tilde{e}^{S_{j}}$$\tilde{e}’$$\mathrm{H}^{3}/G^{S_{j}}$$p^{S_{\mathrm{j}}}$$\mathrm{H}^{3}/G^{S_{j}}arrow \mathrm{H}^{3}/G$$p^{S_{\mathrm{j}}}|E’$$\tilde{e}’$$S^{1}$$\mathrm{H}^{3}/G$$e^{S_{\mathrm{j}}}$$\tilde{e}^{S_{j}}$$\tilde{e}’$$e’$$\mathrm{H}^{3}/G$$e^{S_{j}}$

$e’$

$e’$

$S’\mathrm{x}$$e’$$Sarrow \mathrm{H}^{3}/G$$f_{\dot{1}}$$\mathrm{H}^{3}/G^{S_{j}}$$\tilde{e}’$$S$$S\cross I$$\square$is atotally degenerate -group. By theThus, we have shown thatmain theorem of Bromberg [5], there is asequence of quasi-Fuchsian groups
converging to the isomorphismwith isomorphisms : :
. Let
be a
:
induced by the inclusion of toby the Ahlfors-Bers map.
corresponding topoint inHere,
converges in the Thurston compactification of the Teichmiiller$G^{S_{\mathrm{j}}}$$H_{\dot{1}}$$\iota$$\rho$$\pi_{1}(Sj)arrow H_{\dot{1}}$$\pi_{1}(S_{j})arrow G^{S_{j}}$$S_{j}$$(n_{}^{j},m^{j})$$\mathrm{H}^{3}/G$$(H_{\dot{1}},\rho:)$$\mathcal{T}(S_{j})\cross \mathcal{T}(\overline{S}_{j})$$\{n_{\dot{1}}^{\mathrm{r}j}\}$representing the ending lamination of
space to aprojective lamination, andis constant with respect to .
Recall that we have aparametrization of the quasi-conformal deformais havesincetions of by $qc$ :$[\lambda j]$$e^{S_{j}}$$\sqrt j$$i$$\mathcal{T}(S_{1})\cross\cdots \mathrm{x}\mathcal{T}(S_{k})arrow QH(\Gamma)$$\Gamma$omorphic to($\Gamma\cong\pi_{1}$$\Omega_{\Gamma}/\Gamma$

fromby ahomeomorphism inducing the isomorphism$G\cong\pi_{1}(C)$. We numbered , , so thatfir)/r) to$\partial C$$(\mathrm{H}^{3}\mathrm{U}$$\psi$$S_{1}$$\ldots$$S_{k}$

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DENSITY OF GEOMETRICALLY FINITE KLEINIAN GROUPS,,
face geometrically finite ends. The conformal structure at inof
facing them,finite, that of, determines points
for $j=\kappa+1$ ,, . We define asequence of quasi-conformal deformations
by setting.
Lemma 3. The sequence
converges in $AH(\Gamma)$ , after passing
through a subsequence if necessary.
Proof. This lemma follows from Theorem 3.5 and Lemma 3.6 of Ohshika[13]. We have only to verify that our sequencesatisfies the assumption of Theorem 3.5 using Lernma 3.5 there. Let
be an essential
annulus in. Then, each component ofis contained
either in acomponent ofcorresponding towith $j>\kappa$ on which theconformal structure is constant with respect to , or in acomponent corresponding towith, hence intersects essentially the limit projective
lamination to whichconverges since the lamination is maximal andconnected. By Lemma 3.6 in [13], in the latter case, there exists asequenceof measured laminations
onwith bounded, whose.
limit measured lamination intersects essentially. Therefore, we can apply
Theorem 3.5 in [13] to the quasi-conformal deformations , and see thathas aconvergent subsequence.
Let $(G’, \psi’)\in AH(\Gamma)$ be an algebraic limit of (a subsequence) of.
Lemma 4. The limit group $G’$ has no parabolic elements. Let:
be a homotopy equivalence inducing the isomorphism
:
.
Then,
is homotopic to a homeomorphism fromto.
. We shall prove that $\Psi’|C$ is
Proof. Let $C’$ be acompact core ofhomotopic to ahomeomorphism to $C’$ . This is sufficient to prove the second
sentence of our lemma since the inclusions of both $C$ and $C’$ are homotopic
to homeomorphisms.
Consider acomponentof. If, thenconverges to theprojective laminationas. By the continuity of the length function(see Thurston [21], Ohshika [16], and Brock [3]) together with Theorem 2.2
in Thurston [21], whose proof can be found in [23], as was shown in the
proof of Theorem 8in Ohshika [15], there is asimply-degenerate end ofwhose ending lamination is represented by ameasured lamination
homotopic to, and there is aboundary component of $C’$ homotopic
to.
On the other hand, by Lemma 3in Abikoff [1], for eachwith $j>\kappa$ ,there is ageometrically finite end of
facing aboundary component of$C’$which is homotopic to $\Psi’(Sj)$ . Combining these, we can apply Corollary13.7 in Hempel [8] (originally due to Waldhausen [24]) to conclude that $\Psi’|C$is homotopic to ahomeomorphism to $C’$ . Moreover, since each boundary
component of $C’$ faces aunique end as above, there is no room for asimple
closed curve representing aparabolic element of $G’$ , which must reside on
the boundary of $C’$ .
$S_{\kappa+1}$$\ldots$$S_{k}$$\mathrm{H}^{3}/G$$\ldots$$\Omega_{G}/G$$\mathrm{i}.\mathrm{e}.$$n^{j}\in \mathcal{T}(S_{j})$$k$$\{(\mathrm{r}_{i}, \phi_{i})\}$$(\Gamma_{i}, \phi_{i})=qc(n_{i}^{1}, \ldots, n_{i}^{\kappa}, n^{\kappa+1}, \ldots, n^{k})$$\{(\Gamma_{i}, \phi_{i})\}$$\{(\Gamma_{i}, \phi_{i})\}$$A$$\overline{\mathrm{H}^{3}/\Gamma}=(\mathrm{H}^{3}\cup\Omega_{\Gamma})/\Gamma$$\partial A$$\Omega_{\Gamma}/\Gamma$$S_{j}$$i$$S_{j}$$j\leq\kappa$$\{n_{i}^{j}\}$$\{\lambda_{i}\}$$\partial\overline{\mathrm{H}^{3}/\Gamma}$$(\lambda:)$$1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}_{n_{*}^{\mathrm{j}}}$$A$$\Gamma_{i}$$\{(\Gamma_{i}, \phi_{i})\}$$\square$$\{(\Gamma_{i}, \phi_{\dot{\iota}})\}$$\Psi’$$\mathrm{H}^{3}/G’$$\psi’$$\mathrm{H}^{3}/\Gammaarrow$$\Gammaarrow G’$$\Psi’$$\mathrm{H}^{3}/\Gamma$$\mathrm{H}^{3}/G’$$\mathrm{H}^{3}/G’$$S_{j}$$[\lambda_{j}]$$\partial C$$j\leq\kappa$$\{n_{i}^{j}\}$$iarrow\infty$$\mathrm{H}^{3}/G’$$\Psi’(\lambda_{j})$$\Psi’(S_{j})$$S_{j}$$\mathrm{H}^{3}/G’$$\square$

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KEN’ICHI OHSHIKA

We shall next show that the limit is aquasi-conformal deformation of$G$.

Proposition 5. The limit group $(G’,\psi’)$ is a quasi-conformal deformationof G.
To prove this proposition, we consider first subgroups corresponding tob asubgroup of corresponding. Let
the boundary components of. Then, contains aquasi-conformal deformation:
bytoconverging as $iarrow\infty$ to asubgroup of $(G’,\psi’)$ , which weof.
denote by$\Gamma^{S_{\mathrm{j}}}$$\partial C$$G^{S_{j}}$$\psi$$\Gammaarrow G$$\Gamma$$\mathrm{e}$$\mathrm{r}_{:}$$\Gamma^{S_{\mathrm{j}}}$$(\Gamma_{\dot{1}}^{S_{j}}, \phi_{\dot{1}}^{S_{j}})$$(G^{\prime S_{j}},\psi_{j}’)$Lemma 6. For j.
tion of$\leq\kappa$, the limit groupis a quasi-conformal deforma-$G^{\prime S_{j}}$$G^{S_{\mathrm{j}}}$,such thatcomponentofwhere we regard the first factor as corresponding to the.
Since
acomponent
of
lift
of
ahomeomorphic
that isconverges to the pr0here is equal to , which we defined before,of the Teichmiiller
compactificationin the Thurston
jective lamination. Supstays in acompact set of. We shall show that
space
to which asubsequencepose not. Then, there is aprojective lamination
converges in the Thurston compactification. Since represents anofending lamination, it is maximal and connected on . Hence, by the main
has no convergent
theorem of Ohshika [16], if ( , Aj) $=0$ , then$i(\muj,\lambdaj)>0$.
subsequences. Therefore, we haveor disconnected, eachnotmaximalis eitherOn the other hand, if
boundary component of the minimal hyperbolic subsurface of containing. (This fact isrepresents aparabolic element ofacomponent of,originally due to Thurston [22]. See also Lemma 4.1 in [13].) Since
which is asubgroup of $G’$ , has no parabolic elements, this is impossible.
must also be maximal and connected.
Hence
By Theorem 2.2 in Thurston [21] (and aremark there), there is asesuch thatasconverging toquence of measured laminations
as $iarrow\infty$ . It follows ffom Sullivan’s theorem proved by
Epstein-Marden [7] or Proposition 2.1 in [14] that the length of the realizaalso
by apleated surface inducing the isomorphismtion of ingoes to 0as $iarrow\infty$ . By the continuity of the length function, we see that
. Since $i(\lambda j, \mu j)>0$ , therepresents an ending lamination ofis distinct from the one withend with ending lamination represented by
has two simplyTherefore,.
ending lamination represented bydegenerate ends. As before, using the covering theorem, we get acontradic.
stays in acompact set oftion. Thus we have proved that

Proof There is apair$qc(n_{\dot{1}}’,r_{\dot{l}}’)=(\Gamma_{\dot{1}}^{S_{j}}, \phi_{\dot{l}}^{S_{j}})$$(n_{\dot{l}}’, r_{\dot{1}}’)\in \mathcal{T}(Sj)\mathrm{x}\mathcal{T}(\overline{Sj})$$\Omega_{\Gamma}/\Gamma$$\Omega_{\Gamma}s_{\mathrm{j}}/\Gamma^{S_{\mathrm{j}}}$$n_{\dot{1}}’$$\{n_{\dot{1}}’\}$$n_{\dot{1}}^{j}$$[\lambda_{j}]$$\mathcal{T}(Sj)$$\{r_{\dot{l}}’\}$$\mathcal{T}(S_{j})$$[\mu_{j}]$$\lambda_{j}$$\{r_{\dot{l}}’\}$$S_{j}$$i$$\{(\Gamma_{\dot{1}}^{S_{j}}, \phi_{\dot{1}}^{S_{\mathrm{j}}})\}$$\mu j$$\mu j$$S_{j}$$G^{\prime S_{\mathrm{j}}}$$\mu_{j}$$G^{\prime S_{j}}$$\mu_{j}$$\nu_{k}^{J}$$\mu j$$k$$arrow\infty$$1\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}14’.\cdot(\nu_{\dot{l}}^{j})arrow 0$$\phi_{\dot{1}}^{S_{j}}$$\nu_{\dot{1}}^{j}$$\mathrm{H}^{3}/\mathrm{r}_{:}$$\mathrm{H}^{3}/G^{\prime S_{\mathrm{j}}}$$\mu_{j}$$\mu j$$\lambda j$$\{r_{\dot{l}}’\}$$\mathrm{H}^{3}/G^{S_{\mathrm{j}}}$$\mathcal{T}(Sj)$99
DENSITY OF GEOMETRICALLY FINITE KLEINIAN GROUPS

By taking asubsequence, we can assume thatconverges to apointas. Therefore
quasi-conformal deformais ation ofwith bounded . (Recall that
) This implies thatits limit
is aquasi-conformal deformation of.
$\{r_{i}’\}$$iarrow\infty$$r_{\infty}’\in \mathcal{T}(Sj)$$qc(n_{i}^{j}, m^{j})$$qc(n_{i}’, r_{i}’)$$K_{i}$$K_{i}$$n_{i}’=n_{i}^{j}.$$(G^{;S_{j}}, \psi_{j}’)$$(G^{S_{\mathrm{j}}}, \psi|\Gamma^{S_{j}})$$\square$

Proof of Proposition 5. The lemma above implies that aneighbourhood of

the end of
facingis quasi-isometric to aneighbourhood of
the endby ahomeomorphism in the right homotopy class. Since thisholds for eachfacing asimply-degenerate end, this means that there isaquasi-isometry in the right homotopy class from;henceto$(G’, \psi’)$ is aquasi-conformal$[]$
deformation of $(G, \psi)$ .
$\mathrm{H}^{3}/G’$$\Psi’(S_{j})$$e^{S_{j}}$$S_{j}$$\mathrm{H}^{3}/G$Let:$\mathrm{H}^{3}/G’$

be aquasi-conformal homeomorphism such that
corresponding to
the geometrically finite ends are the same as those on
by our definition of , this map is taken to be conformal on. By Sullivan’s rigidity
theorem ([18]), this implies that is in fact aconformal homeomorphism;
hence $(G, \psi)=(G’, \psi’)$ . Thus, we have proved that $(G, \psi)$ is also alimit ofgeometrically finite groups, and completed the proof of Theorem 1.
$f$$S_{\infty}^{2}arrow S_{\infty}^{2}$$f\psi f^{-1}=\psi’$. Since the conformal structures of$\Omega c’/G’$$\Omega c/G$$\Gamma_{i}$$f$$\Omega_{G}$$f$$(\Gamma_{i}, \phi_{i})$REFERENCES[1] W. Abikoff, On boundaries of Teichmller spaces and on Kleinian groups, III, Acta
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$3\sim$$\mathrm{G}\mathrm{T}/9801045$$\mathrm{G}\mathrm{T}/9801019$

DEpARTMENT OF MATIIEMATICS, GRADUATE SCHOOL OF SCIENCE, OSAKA UNIVERSITY, TOYONAKA, Osaka 560-0043, iApAN