数理解析研究所講究録
第 1571 巻 2007 年 1-4

1

DIVERGENCE IN DEFORMATION SPACES OFKLEINIAN GROUPS
KEN’ICHI OHSHIKA
DEPARTMENT OF MATHEMATICS, OSAKA UNIVERSITYThis note is based on my talk given in the RIMS on the 4th ofDecember, 2006.
The attention of specialists in the Kleinian group theory is now
shifted to the study of the topological structure of deformation spaces
ffier the major problems like Maxden’s tameness conjecture and theending lamination conjecture $axe$ solved. Although we know, by the resolution of the Bers-Thurston density conjecture ([4]) using the proofof the ending lanination conjecture by Minsky with his collaboratorsthat every finitely generated Kleinian group is an algebraic hmit ofquasi-conformal deformations of a(minimally parabolic) geometricany
finite group, the structure of deformation spaces as topological spaces
is far ffom completely understood.
To understtd sucha global structure of deformation spaces, thefirst step would be to give acriterion for sequences in the deformation
space to converge or diverge. Let us put it in more concrete tems focusing only on the case of Kleinian groups isomorphic to surface groups.
the space of faithfulConsider ahyperbolic surface of ffiite typepreserving the parabolicity
to&screte representations ofand complex conjugamodulo conjugacy(both as elements oftion), whii is usually denoted by $AH$ (-S). Since the hyperbolic metric,
toof determines aFuchsian representation ofas the space of quasi-conformal deformations of tbis representation, weconsider the space of quasi-Kchsit representations $QF(S)$ embedded as open set in $AH(S)$ . What we are interested in is the problem
to determine in which directions $QF(S)$ has bontier in $AH(S)_{\bm{t}}d$ inwhich directions it is open-ended. Since by the theory of Ahlfors-Bers,, we can describe the directions$QF(S)$ is parametrised by
in $QF(S)$ in terms of the Teichm\"uller spaces.
results in this talk is the following.
The$\bm{t}d$$S$$\pi_{1}(S)$$PSL_{2}\mathbb{C}$$PSL_{2}\mathbb{C}$$\pi_{1}(S)$$S$$PSL_{2}\mathbb{R}\subset PSL_{2}\mathbb{C}$$C\bm{t}$$\bm{t}$$\mathcal{T}(S)\cross \mathcal{T}(\overline{S})$$ma\dot{i}$Theorem 1. Let
the following conditions.

$\{(m_{i},n_{i})\}$

be a sequence in$\mathcal{T}(S)\cross \mathcal{T}(\overline{S})$satisfy ing

2KEN’ICHI OHSHIKA

converges to a projective lamination.
converges towhereas
which is notand
share a component(2) The supports ofa simple closed curve.
Then the sequence $\{qf(m_{i}, n_{i})\}\subset QF(S)$ diverges in $AH(S)$ .
(1)$[\mu^{-}]\in \mathcal{P}\mathcal{M}\mathcal{L}(S)$$\{m_{i}\}$$[\mu^{+}]\in \mathcal{P}\mathcal{M}\mathcal{L}(S)$$\{n_{i}\}$$\mu^{+}$$\mu^{-}$$\mu_{0}$be two measured laminations on suchandTheorem 2. Letthat the components shared by $|\mu^{-}|and|\mu^{+}|$ are all simple closed curves,.
which we denote bylie on the $bounda\eta$ of support(1) Suppose that none ofor . Then there is a seing surfaces of components ofwith convergent $qf(m_{i},n_{i})$inquence
converges
and converges tosuch that. Moreover, if $|\mu^{+}|=c_{1}\cup\cdots\cup c_{r}$ , we chooseso that $qf(m_{i}, n_{i})$ converges exotically to a b-group.
convergconverging toand(2) Otherwise for every, the sequence $\{qf(m_{i}, n_{i})\}\subset QF(S)$ diverges ining to$S$$\mu^{+}$$\mu^{-}$$c_{1},$$\ldots,$$c_{r}$$c_{1},$$\ldots,$$c_{r}$$\mu^{+}$$\mu^{-}$$\{(m_{i},n_{i})\}$$\mathcal{T}(S)\cross \mathcal{T}(\overline{S})$$[\overline{\mu}^{-}]$$m_{i}$$|\mu^{-}|,$$[\overline{\mu}^{+}]and|\overline{\mu}^{-}|=$$n_{i}$$|\overline{\mu}^{+}|=|\mu^{+}|$$\{(m_{\dot{j}}, n_{i})\}$$\{n_{i}\}$$[\mu^{-}]$$\{m_{i}\}$$[\mu^{+}]$$AH(S)$ .

The proofs of Theorem 1 and Theorem 2 take quite different strategies. For Theoren 1, which is apparently the more complicated caseof the two, we can use a rather standard technique of pleated surfaces

originally due to Thurston. For Theorem 2, we need to invoke much
more sophisticated tool of model manifolds due to Minsky.
In this note we only explain Theorem 1.
1. A SKETCH OF PROOF OF THEOREM 1.
Let be a hyperbolic surface of finite area. Let :be a quasi-Fttchsian representation representing $qf(m_{i}, n_{i})$ as was given
the hyperbolic 3be the image of , andin Theorem 1. Let. Since
is a quasi-conformal deformation of themanifold
associated to the hyperbolic metricFuchsian representation of:
inducedon , there is a natural homeomorphism
as theby a quasi-conformal homeomorphism, where we regard
hyperbolic 3-manifold containing the hyperbolic surface in the formof
as a totally geodesic submanifold. Since is quasi-FUchsian,is geometrically finite and has convex core $C(M_{i})$ ,the manifoldpreserving the parabolicity. We canwhich is homeomorphic to$\Phi_{i}(S\cross[-1,1])=C(M_{i})$.
isotopeabove so that
$C(M_{i})$components
correspondingof
be the two frontierLet$\Phi_{i}(S\cross\{-1\})$
respectively.
hyperbolic
metricTheandtoand
as length metrics.
induces hyperbolic structures onon$S$$\phi_{i}$$G_{i}$$\mathbb{H}^{3}/G_{i}$$\pi_{1}(S)arrow PSL_{2}\mathbb{C}$$M_{i}$$\phi_{i}$$G_{i}$$\pi_{1}(S)$$S$$\Phi_{i}$$S\cross \mathbb{R}arrow M_{i}$$S\cross \mathbb{R}$$S$$G_{i}$$S\cross\{0\}$$M_{i}$$S\cross I$$\Phi_{i}$$\Sigma_{i}^{-},$$\Sigma_{i}^{+}$$\Phi_{i}(S\cross\{1\})$$M_{2}$$\Sigma_{i}^{-}$$\Sigma_{i}^{+}$

3DIVERGENCE IN DEFORMATION SPACES OF KLEINIAN GROUPSWe give markings onby natural homeomorphism between
and$S\cross\{-1\}$andandobtained by forgetting the second coordiis different $hom$nates. It should be noted the orientation given on
the ordinry one induced $homC(M_{i})$ . Let
be points In
determined by these hyperbolic structures onand markings.
Since $(G_{i}, \phi_{i})=qf(m_{i}, n_{i})$ with respect to the Ahlfors-Bers parametrisation, by Bers’ inequality, there is auniversal bound $K$ between theTeichm\"uUer dlstances between.
andgive two measured laminations
The pleating loci ontd.
on by pulling ba&them to using the inverse ofand
By passing to asubsequence, we can assume that both$aJso$
converge to projective laminations
and. We canassumeconverge
sequences
geodesic
tosupportsthat theofandlaminationsin the Hausdorff topoloy.
We shall prove Theorem 1by contradiction. Assume thatconverges toin $AH(S)$ by taking conjugates and asubsequence.
We divide our argument into three cases:(1) The first case is when eitheror
non-zero.
(2) The second case is when both
acomponentcontainandbyasimple
sharedcloeed curve.
which is not(3) Finally, the third case is when either
is disjoint $hom$orcomponent ofthat is not asimple closed
shared withcurve.
. The argument
In the first case, we assume that$i(\mu^{+},\lambda^{+})>0$
for the case whenis completely the same. By thedefinition of the Thurston compactification of the Teichm\"uller space(see $Fathi- Laudenbach- Po\acute{e}naru[2]$ ) or the argument in Otal [5], we. Since is realised on , its length on
have.
. Therefore we havewith respect tois equal to that in.
. On the other hand, by the continuity of thelength function (Brock [1]), we have$\Sigma_{i}^{+}$$\Sigma_{i}^{-}$$S$$S\cross\{1\}$$\Sigma^{+}$$(p_{i}, q_{i})$$\Sigma_{i}^{-},$$m_{i},p_{i}$$\Sigma_{i}^{-}$$\lambda_{i}^{-},$$S$$\lambda_{i}^{+}$$n_{i},$$\mathcal{T}(S)$$\Sigma_{i}^{+}$$q_{i}$$\Sigma_{i}^{+}$$S$$\Phi_{i}|S\cross\{\pm 1\}$$[\lambda_{i}^{-}]$$[\lambda_{i}^{+}]$$[\lambda_{\infty}^{+}]$$[\lambda_{\infty}^{-}]$$\{|\lambda_{i}^{+}|\}$$\{|\lambda_{i}^{-}|\}$$\ell_{\infty}^{-}\bm{t}d\ell_{\infty}^{+}$$\{(G_{i}, [\phi_{i}])\}$$(\Gamma, \psi)$$i(\mu^{+}, \lambda_{\infty}^{+})is$$i(\mu^{-}, \lambda_{\infty}^{-})$$\lambda_{\infty}^{+}$$\lambda_{\infty}^{-}$$\mu^{\mp}$$\lambda_{\infty}^{+}$$\mu^{+}$$\bm{r}y$$\lambda_{\infty}^{-}$$\mu^{-}$$i(\mu^{-}, \lambda_{\infty}^{-})>0$$1engh_{\Sigma^{-}}$ $(\lambda_{i}^{-})arrow\infty$$\Sigma_{i}^{-}$$\Sigma^{-}$$\lambda_{j}^{-}$$M_{i}$$p_{i}$$1engh_{M}:(\Phi_{i}(\lambda_{i}^{-}))arrow\infty$$\lim 1ength_{M:}(\Phi_{i}(\lambda_{i}^{-}))=1engh_{N}(\Psi(\lambda_{\infty}^{-}))$and the right hand side is finite. This is a contradiction, and we havecompleted the proof of the first case.
be the component shared
Now let us turn to the second case. Let, which is not a simple closed curve.
by IandUsing the technique of interpolating pleated surfaces due to Thurston,
we prove the following.
$\lambda_{0}$$\lambda_{\infty}^{+}|$$|\lambda_{\infty}^{-}|$Proposition 3. We can take a constant $L>0$ for which the following
is homotopic
holds for large . There is $t_{i}\in[0,1]$ such that$i$

$H_{i}(S(\mu_{0}),t_{i})$4KEN’ICHI OHSHIKAtoby a homotopy staying within the distance fromwhich keeps the frontier inside the Margulis tubes all the time.
$L$$f_{i}|S(\mu_{0})$$f_{i}(S(\mu_{0}))$converges to a pleated surface
Then the pleated surface
homotopic to:since the homotopy between
has bounded diameter and converges to a homoand. The limit pleated surface realises
andtopy between. By taking a subsethe limit of the measured laminations
converges to a projective laminationquence we can assume that, which must have the same support asif it restrictedon. Therefore the limit pleated surface realises . Since isin
is also lifted to a pleatedlifted to $f’$ : $Sarrow N$ , the pleated surface
repsurface, which also realises . This contradicts the fact that
resents an ending lamination. Thus we have completed the proof ofTheorem 1 in this case.
The third case is the most difficult. We need to make an eclectic
approach considering Hausdorff limits of the bending loci. The key
steps are as follows.
$g_{i}|S(\mu_{0})$$S(\mu_{0})arrow M_{\infty}$$g_{\infty}$$g_{i}|S(\mu_{0})$$f_{\infty}$$f_{i}|S(\mu_{0})$$g_{\infty}$$f_{\infty}|S(\mu_{0})$$g_{\infty}$$\alpha_{i}(t_{i})|S(\mu_{0})$$\alpha(t_{i})$$\alpha([0,1])$$lis$$\mu_{0}$$S(\mu_{0})$$f_{\infty}$$\mu_{0}$$g_{\infty}$$\mu_{0}$$\mu_{0}$Lemma 4. Let be a minimal component of
not intersect a component of transversely.
$\ell$

or$\ell_{\infty}^{-}$$\ell_{\infty}^{+},$

Then$\ell$

does

$\mu$

contain aLemma 5. Suppose that the Hausdorff limitsof I
common component which coincides with the support a component
connecting
Utthof . Then there is an arc :such that for any sequence
converging uniformly to an arc
converges inin $[0,1]$ and monotone increasing
for whichthe Hausdorff topology, the limit contains a minimat component which
except for the case whenor $t_{k}=1-1/4i_{k}$coincides with.
orfor all large , in which case we have$\ell_{\infty}^{\mp}$$\lambda_{i}^{\mp}|$$\mu_{0}$$\mu^{\mp}$$\alpha_{i}$$[\lambda_{i}^{+}]$$[0,1]arrow \mathcal{P}\mathcal{M}\mathcal{L}(S)$$[\lambda_{i}^{-}]$$\{t_{k}\}$$\alpha_{\infty}$$\{i_{k}\}$$|\alpha_{i_{k}}(t_{k})|$$t_{k}=1/4i_{\dot{k}}$$|\mu_{0}|$$k$$[\alpha_{i_{k}}(t_{k})]=[\lambda_{\infty}^{+}]$$[\lambda_{\infty}^{-}]$REFERENCES[1] J. Brock, Continuity of Thurston’s length function, Geom. Funct. Anal. 10,(2000), 741-797.
[2] A. Fathi, V. Po\’enaru, et F. Laudenbach, Travaux de Thurston sur les surfaces,
S\’eminaire Orsay, Ast\’erisque 66-67, (1979).
[3] K. Ohshika, Divergent sequences of Kleinian groups, The Epstein birthday. Topol. , Univ. Warwick, Coventry,
schrift $G\infty m$ . Topol. Monogr, 1,$G\infty m$(1998), 419-450.
[4] K. Ohshika, Realising end invariants by limits of minimally parabolic groups,

arXiv: math. $GT/0504546$[5] J-P. Otal, Le th\’eor\‘eme d’hyperbolisation pour les vari\’et\’es fibrdes de dimension
3, Ast\’erisque 235 (1996).