Ending laminations in the Masur domain.

*(English)*Zbl 1052.57020
Komori, Y. (ed.) et al., Kleinian groups and hyperbolic 3-manifolds. Proceedings of the Warwick workshop, Warwick, UK, September 11–14, 2001. Cambridge: Cambridge University Press (ISBN 0-521-54013-5/pbk). Lond. Math. Soc. Lect. Note Ser. 299, 105-129 (2003).

In this paper the authors study the relation between the geometry and the topology of the ends of a hyperbolic 3-manifold \(M\) whose fundamental group is not a free group. In the Introduction they state the two main Theorems and they also describe the strategy of the proof of the first one. This Theorem is a converse to a result by R. D. Canary [J. Am. Math. Soc. 6, No. 1, 1–35 (1993; Zbl 0810.57006)], in the case where \(\pi_1(M)\) is not free. First Theorem: Let \(E\) be a compressible end of a complete oriented hyperbolic 3-manifold \(M\) whose fundamental group is finitely generated but not free. If a Masur domain lamination on \(\partial E\) is not realized in \(M\), the end \(E\) is tame.

In order to state the second Theorem we need a bit of notation. Let \(N\) be a non-trivial compression body, \(\rho_0 : \pi_1(N) \rightarrow \text{PSL}_2 \mathbb{C}\) a geometrically finite and minimally parabolic representation which uniformizes \(N\) and let \(QH(\rho_0)\) be the space of conjugacy classes of quasi-conformal deformations of \(\rho_0\). Second Theorem: Let \(N\) be a compression body which is not a handlebody and let \((\rho_i)\subset QH(\rho_0)\) be a sequence which converges algebraically to \(\rho : \pi(N) \rightarrow \text{PSL}_2 \mathbb{C}\). If \((\rho_i)\) converges into the Masur domain, then \(M_\rho = \mathbb{H}^3/\rho(\pi_1(N))\) is tame.

The authors give a long section of preliminaries about compact cores, compression bodies, function groups, boundary groups, laminations on surfaces and pleated surfaces. Two sections follow entitled “Laminations on the exterior boundary” and “Compactness theorem”. In the second one, the main result is the following Proposition: A lamination \(\lambda \in \mathcal{O}_P\) is realized in \(M_P\) either if \(\lambda\) is not minimal arational or if \(\lambda\) is minimal arational and there is a sequence \((\gamma_i)\) of multicurves converging to \(\lambda\) and a compact set \(K \subset M_P\) such that \(\gamma_i\) is realized by a pleated surface \(f_i : X_i \rightarrow M_P\) with \(f_i(X_i)\cap K\neq\emptyset\) for all \(i\).

In the last section the proofs of both Theorems are given. Finally, the authors prove the following Proposition: For every compression body \(N\) and every Masur domain lamination \(\mu\) on \(\partial_\epsilon N\) there is a discrete and faithful representation \(\rho : \pi_1(N) \rightarrow \text{PSL}_2 \mathbb{C}\) such that no component of \(\mu\) is realized in \(M_P\).

For the entire collection see [Zbl 1031.30002].

In order to state the second Theorem we need a bit of notation. Let \(N\) be a non-trivial compression body, \(\rho_0 : \pi_1(N) \rightarrow \text{PSL}_2 \mathbb{C}\) a geometrically finite and minimally parabolic representation which uniformizes \(N\) and let \(QH(\rho_0)\) be the space of conjugacy classes of quasi-conformal deformations of \(\rho_0\). Second Theorem: Let \(N\) be a compression body which is not a handlebody and let \((\rho_i)\subset QH(\rho_0)\) be a sequence which converges algebraically to \(\rho : \pi(N) \rightarrow \text{PSL}_2 \mathbb{C}\). If \((\rho_i)\) converges into the Masur domain, then \(M_\rho = \mathbb{H}^3/\rho(\pi_1(N))\) is tame.

The authors give a long section of preliminaries about compact cores, compression bodies, function groups, boundary groups, laminations on surfaces and pleated surfaces. Two sections follow entitled “Laminations on the exterior boundary” and “Compactness theorem”. In the second one, the main result is the following Proposition: A lamination \(\lambda \in \mathcal{O}_P\) is realized in \(M_P\) either if \(\lambda\) is not minimal arational or if \(\lambda\) is minimal arational and there is a sequence \((\gamma_i)\) of multicurves converging to \(\lambda\) and a compact set \(K \subset M_P\) such that \(\gamma_i\) is realized by a pleated surface \(f_i : X_i \rightarrow M_P\) with \(f_i(X_i)\cap K\neq\emptyset\) for all \(i\).

In the last section the proofs of both Theorems are given. Finally, the authors prove the following Proposition: For every compression body \(N\) and every Masur domain lamination \(\mu\) on \(\partial_\epsilon N\) there is a discrete and faithful representation \(\rho : \pi_1(N) \rightarrow \text{PSL}_2 \mathbb{C}\) such that no component of \(\mu\) is realized in \(M_P\).

For the entire collection see [Zbl 1031.30002].

Reviewer: Ernesto MartĂnez (Madrid)