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On the degeneration of Schottky groups. (Sur la dégénérescence des groupes de Schottky.) (French) Zbl 0828.57008
Suppose $$\lambda$$ and $$\lambda'$$ are two measured laminations which fill a hyperbolic surface $$S$$ (this means that a lift of any component of $$S- (\lambda \cup \lambda')$$ to $$\text{H}^2$$ is either bounded or lies in a horoball region of a cusp point). The Double Limit Theorem of Thurston says that given a sequence $$(\rho_i : \pi_1 (S) \to \text{PSL} (2,C))$$ of quasifuchsian groups, if the corresponding sequences of hyperbolic structures on the quotient surfaces of the two components of the region of discontinuity converge to $$\lambda$$ and $$\lambda'$$, then the sequence $$(\rho_i)$$ has a convergent subsequence. Thurston has conjectured an analogue for Schottky groups, i.e. geometrically finite structures on a 3-dimensional handlebody $$H$$. Consider pairs $$(\rho, \varphi)$$, where $$\rho : G = \pi_1 (H) \to \text{PSL} (2,C)$$ and $$\varphi$$ is a diffeomorphism from $$H$$ to $$(\text{H}^3 \cup \Omega (\rho (G)))/o(G)$$. By the Ahlfors-Bers Theorem the space of such structures is homeomorphic to the Teichmüller space $${\mathcal T} (\partial H)$$. Let $$\lambda$$ be a geodesic lamination in the Thurston boundary of $${\mathcal T} (\partial H)$$, and assume that $$\lambda$$ lies in the Masur domain (a certain open subset of $${\mathcal T} (\partial H)$$ on which the mapping class group of $$H$$ acts properly discontinuously). The conjecture is that given a sequence of geometrically finite structures $$((\rho_i, \varphi_i))$$, if the corresponding hyperbolic structures on $$\partial H$$ converge to $$\lambda$$, then $$(\rho_i)$$ has a convergent subsequence. In this paper, the author proves Thurston’s conjecture for the case when $$H$$ has genus 2 and each component of $$\partial H - \lambda$$ is simply-connected. This overlaps partially with cases proven by R. Canary.
The author’s work is in the context of a more general conjecture, motivated as follows. Suppose that $$(\rho_i)$$ has no convergent subsequence. By a result of J. Morgan and P. Shalen, one can obtain an action of $$G$$ on an R-tree $$T$$ such that the stabilizer of any noncyclic vertex is cyclic (i.e. “the action has small stabilizers”), and such that for some subsequence of $$(\rho_i)$$, the ratio of the translation lengths of any two elements on $$G$$ (for which the denominator does not have translation length 0) is the limit of the ratios of their translation lengths as elements of the $$\rho_i (G))$$. There is a $$G$$- equivariant map $$F$$ (well-defined up to $$G$$-equivariant homotopy) from the universal cover $$\text{H}^3$$ of $$H$$ to $$T$$. Let $$P^1 (\text{H}^3)$$ be the projective unit tangent bundle of $$\text{H}^3$$, and let $$P : P^1 (\text{H}^3) \to \text{H}^3$$ be the projection. A closed subset $$X \subset P^1 (\text{H}^3)$$, invariant under the geodesic flow, is said to be realized in $$T$$ if the restriction of $$F \circ P$$ to $$X$$ is $$G$$-equivariantly homotopic to a map which carries each geodesic in $$X$$ injectively into $$T$$. The general conjecture is that for any action of $$G$$ on an $$R$$-tree $$T$$, with small stabilizers, every measured lamination in the Masur domain is realized in $$T$$.
The author proves the conjecture for minimal laminations when the genus of $$H$$ is 2 or (more generally) when the action of $$G$$ on $$T$$ is dual to a measured lamination in a compact surface with boundary. This implies the case of Thurston’s conjecture given above. The proofs use a variety of geometric methods, together with results of R. Skora, and M. Culler and K. Vogtmann.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 30F60 Teichmüller theory for Riemann surfaces 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010)
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