Convergence of freely decomposable Kleinian groups

Abstract

We consider a compact orientable hyperbolic 3-manifold with a compressible boundary. Suppose that we are given a sequence of geometrically finite hyperbolic metrics whose conformal boundary structures at infinity diverge to a projective lamination. We prove that if this limit projective lamination is doubly incompressible, then the sequence has compact closure in the deformation space. As a consequence we generalise Thurston’s double limit theorem and solve his conjecture on convergence of function groups affirmatively.

Appendix

For the sake of completeness, we shall give brief proofs of some propositions which we cited in previous sections.

Proposition 2.2

[23], Sect. 2 paragraphs after Lemme 2.7] Let $\Sigma$ and $\Sigma '\subset \partial _{\chi <0} M$ be two compact, connected, incompressible surfaces which are disjoint or equal and do not contain any essential closed curve which can be homotoped into $\partial _{\chi =0} M$. Let $\tilde{\Sigma }\subset \partial \tilde{M}$ (resp. $\tilde{\Sigma }'$) be a connected component of the preimage of $\Sigma$ (resp. $\Sigma '$) and let $\Gamma \subset \rho (\pi _1(M))$ (resp. $\Gamma '$) be the stabiliser of $\tilde{\Sigma }$ (resp. $\Gamma '$).

Then $\overline{\tilde{\Sigma }} \cap \overline{\tilde{\Sigma }}'$ is either empty or equal to the limit set of $\Gamma \cap \Gamma '$.

In the latter case, if $\Gamma \cap \Gamma '$ is not cyclic, then it is the fundamental group of a (possibly twisted) I-bundle which is a connected component of a characteristic submanifold of $(M,\Sigma \cup \Sigma ')$. If $\Gamma \cap \Gamma '$ is cyclic, then it is a finite index subgroup of a solid torus which is a connected component of a characteristic submanifold of $(M,\Sigma \cup \Sigma ')$.

Proof

Suppose that $\xi$ belongs to the limit set of $\Gamma \,\cap \,\Gamma '$. Since we are considering only geometrically finite groups, both $\Gamma$ and $\Gamma '$ are convex cocompact. Let l be a geodesic ray from the origin $O\in {\mathbb H}^3$ to $\xi$. Let F be a fundamental domain of the convex core of $\Gamma$ containing O, and $F'$ a fundamental domain of the convex core of $\Gamma '$ containing O. Since $\Gamma$ and $\Gamma '$ are convex cocompact, the diameters of F and $F'$ are bounded. Choose $g_n\in \Gamma , g_n'\in \Gamma '$ such that $g_n F\cap l\ne \emptyset , g_n'F'\cap l\ne \emptyset$ and $g_nF\cap g_n'F' (\ne \emptyset ) \longrightarrow \xi$. Since the diameters of F and $F'$ are bounded, we have

$$\begin{aligned} d(g_nO, g_n'O)=d(O,g_n^{-1}g_n' O)\le K \end{aligned}$$

for all n and a fixed K. By discreteness of $\rho (\pi _1(M))$, after passing to a subsequence, $g_n^{-1}g_n'=g$ for all n. Then $g_n^{-1}g_n'=g_m^{-1}g_m'$, i.e., $g_m g_n^{-1}=g_m'g_n'^{-1}\in \Gamma \cap \Gamma '$. Since $\Gamma$ and $\Gamma '$ are convex cocompact, these elements are hyperbolic and the limit set of $\Gamma \cap \Gamma '$ contains at least two points. Let $h\in \Gamma \cap \Gamma '$ be a hyperbolic element. Then the invariant geodesics $\tilde{c}, \tilde{c}'$ of h in $\tilde{\Sigma },\tilde{\Sigma }'$ descend to closed geodesics c and $c'$ in $\Sigma$ and $\Sigma '$. Hence c and $c'$ bound an annulus A (not necessarily embedded) which is not homotoped into $\partial M$. Hence A must be contained in some characteristic submanifold of $(M,\Sigma \cup \Sigma ')$. If $\Sigma =\Sigma '$, then $c=c'$ and A is a Möbius band.

Let $F=\overline{\tilde{\Sigma }} \cap \overline{\tilde{\Sigma }}'$ and $C,C'$ be the projections of the convex hulls of F in $\tilde{\Sigma }$ and $\tilde{\Sigma }'$. Then $\pi _1(C)=\Gamma \cap \Gamma '$ and $C\cup C'$ is the boundary of an I-bundle $C\times I$ which is essential in $(M,\Sigma \cup \Sigma ')$. If $\Sigma =\Sigma '$, then it is a twisted I-bundle with the boundary C. $\square$

Next we summarise some proofs from [22] for the reader’s convenience.

Proposition 8.1

Suppose that the action of $\pi _1(M)$ on an ${\mathbb R}$-tree $\mathcal T$ is minimal and small. Let $S\subset \partial M$ be a compact compressible surface which has (possibly empty) geodesic boundary with respect to a hyperbolic metric on $\partial M$. Suppose that $\phi :\mathcal T_\mu {\rightarrow }\mathcal T$ is a $\pi _1(M)$-equivariant morphism which folds only at complementary regions. If $\mu \subset S$ is in tight position with respect to a meridian $m\subset S$, then $|\mu |$ can be extended to a geodesic lamination which contains a homoclinic leaf h in S.

Proof

Fix a hyperbolic metric on $\partial M$. The universal cover $\tilde{S}\subset {\mathbb H}^2$ is a convex subset of ${\mathbb H}^2$. Let $\bar{m}$ represent an element of $\pi _1(S)$ corresponding to the meridian m, which leaves invariant a lift $\tilde{m}\subset \tilde{S}$. By equivariance, for $x\in \tilde{m}$,

$$\begin{aligned} \phi (\pi (\bar{m} x))=\phi (\pi (x)) \end{aligned}$$

since $\bar{m}$ is trivial in $\pi _1(M)$, where $\pi :\tilde{S}{\rightarrow }\mathcal T_\mu$ is an equivariant map. Since $\phi$ folds $[\pi (x), \pi (\bar{m} x)]$ only finitely many times, one can find segments $\tilde{I}_1, \tilde{I}_2 \subset \tilde{m}$ such that $\tilde{I}_1\cap \tilde{I}_2=y\in \tilde{m}$ and $\phi$ folds $\pi (\tilde{I}_1)$ and $\pi (\tilde{I}_2)$ along $\pi (y)$. For $x\in m\cap \mu$, let $\mu _x^+$ denote a half leaf of $\mu$ starting from x to a chosen positive direction. Then using Skora’s idea, Kleineidam and Souto showed [22] (Proposition 3) that there are $z_i\in I_i\cap \mu$ such that the lifts of $\mu _{z_1}^+$ and $\mu _{z_2}^+$ to $\partial \tilde{M}$ have the same endpoints. Let C be the complementary region of $\tilde{\mu }$ in $\tilde{S}$, which contains the folding point y. Let $\tilde{\mu }_1,\tilde{\mu }_2$ be boundary leaves of C. Up to reversing the orientation, we can assume that $\tilde{\mu }_1^+,\tilde{\mu }_2^+$ are not asymptotic in $\tilde{S}$. Since the lifts of $\mu _{z_1}^+$ and $\mu _{z_2}^+$ to $\partial \tilde{M}$ have the same end points, by shrinking the intervals, we can see that the projection of $\tilde{\mu }_1^+,\tilde{\mu }_2^+$ to $\partial \tilde{M}$ have the same endpoint. Let l be the geodesic in $\tilde{S}$ joining the end points of $\tilde{\mu }_1^+,\tilde{\mu }_2^+$. The projection of l to S becomes a homoclinic leaf disjoint from $\mu$. $\square$

Proposition 8.2

Let S be a compressible surface in $\partial M$, which contains a homoclinic leaf h. Then there is a sequence of meridians whose Hausdorff limit does not cross h.

Proof

Since a homoclinic leaf cannot be contained in an incompressible surface by Lemma 2.1, $S(\bar{h})$ must contain a meridian m.

If h contains infinitely many homotopy classes of m-waves. Then there are $(x_i),(y_i)\subset {\mathbb R}$ such that $h(x_i),h(y_i)\in m$ and $h[x_i,y_i]$ are non-homotopic m-waves. Since m is compact, after passing to a subsequence, we may assume that $h(x_i)$ and $h(y_i)$ converge. Hence for any $\epsilon >0$, we can choose i, j such that the lengths of segments $[h(x_i),h(x_j)],[h(y_i),h(y_j)]\subset m$ are less than $\epsilon$. Then $h[x_i,y_i]\cup h[x_j,y_j]\cup [h(x_i),h(x_j)]\cup [h(y_i),h(y_j)]$ is a meridian whose geodesic representative lies nearby the homoclinic leaf.

If h contains only finitely many homotopy classes of m-waves. then there is a meridian m and two disjoint half-leaves $h^+$ and $h^-$ of h such that $h^+$ and $h^-$ are in tight position with respect to m. Considering the intersections of m and $h^+$ and $h^-$ respectively, one obtains a picture similar to . 8, namely there is an arc $k\subset h^+$ and an arc $k'\subset h^-$ which nearly bounds an annulus and a wave between k and $k'$. These arcs can be used to construct a sequence of meridians whose Hausdorff limit does not cross h as explained in the proof of [22, Proposition 1]. $\square$

With more work, one could probably prove that $\bar{h}$ is a Hausdorff limit of meridians. On the other hand, in all the situations we have used Proposition 8.2, with only little changes, we could have replaced it with the following weaker result whose proof is easier.

Lemma 8.3

Let S be a compressible surface which contains a homoclinic leaf h, and let $\beta$ be a measured lamination which does not cross h. Then there is a sequence of meridians whose Hausdorff limit does not cross $\beta$.

Proof

Using cut-and-paste operation as in Lemma 2.3, we construct a sequence of meridians $m_i$ such that $i(m_i,\beta )\longrightarrow 0$. Start with a meridian $m\subset S(\bar{h})$. Since h is homoclinic, it contains an m-wave. Using this wave as in the proof of Lemma 2.3, we get a meridian $m_1$ such that $i(m_1,\beta )\le \frac{1}{2} i(m,\beta )$. Then we do the same again on $m_1$. Repeating this, we get a sequence of meridians $m_i$ such that $i(m_i,\beta )\le \frac{1}{2^i} i(m,\beta )$. $\square$