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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.

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)
Full Text: DOI
[1] M. Bestvina, Degenerations of the hyperbolic space , Duke Math. J. 56 (1988), no. 1, 143-161. · Zbl 0652.57009
[2] F. Bonahon, Bouts des variétés hyperboliques de dimension \(3\) , Ann. of Math. (2) 124 (1986), no. 1, 71-158. JSTOR: · Zbl 0671.57008
[3] R. D. Canary, Algebraic convergence of Schottky groups , Trans. Amer. Math. Soc. 337 (1993), no. 1, 235-258. JSTOR: · Zbl 0772.30037
[4] R. D. Canary, D. B. A. Epstein, and P. Green, Notes on notes of Thurston , Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3-92. · Zbl 0612.57009
[5] M. Culler and K. Vogtmann, The boundary of outer space in rank two , Arboreal group theory (Berkeley, CA, 1988), Math. Sci. Res. Inst. Publ., vol. 19, Springer, New York, 1991, pp. 189-230. · Zbl 0786.57002
[6] A. Fathi, F. Laudenbach, and V. Poenaru, Travaux de Thurston sur les surfaces , Astérisque, vol. 66, Société Mathématique de France, Paris, 1979. · Zbl 0406.00016
[7] M. Gromov, Three remarks on geodesic dynamics and fundamental group , prépublication. · Zbl 1002.53028
[8] M. Gromov, Hyperbolic groups , Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75-263. · Zbl 0634.20015
[9] H. Masur, Measured foliations and handlebodies , Ergodic Theory Dynam. Systems 6 (1986), no. 1, 99-116. · Zbl 0628.57010
[10] J. W. Morgan and J.-P. Otal, Relative growth rates of closed geodesics on a surface under varying hyperbolic structures , Comment. Math. Helv. 68 (1993), no. 2, 171-208. · Zbl 0795.57009
[11] J. W. Morgan and P. B. Shalen, Valuations, trees, and degenerations of hyperbolic structures. I , Ann. of Math. (2) 120 (1984), no. 3, 401-476. JSTOR: · Zbl 0583.57005
[12] J. W. Morgan and P. B. Shalen, Degenerations of hyperbolic structures. III. Actions of \(3\)-manifold groups on trees and Thurston’s compactness theorem , Ann. of Math. (2) 127 (1988), no. 3, 457-519. JSTOR: · Zbl 0661.57004
[13] J.-P. Otal, Courants géodesiques et produits libres , Thése d’Etat, Orsay, 1989, article en préparation.
[14] J.-P. Otal, Le théorème d’hyperbolisation pour les variétés fibrées de dimension trois , prépublication no 125, E.N.S. Lyon, 1994.
[15] F. Paulin, Topologie de Gromov équivariante, structures hyperboliques et arbres réels , Invent. Math. 94 (1988), no. 1, 53-80. · Zbl 0673.57034
[16] R. Skora, Surface groups acting on \(\mathbfR\)-trees , prépublication.
[17] W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces , Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. · Zbl 0674.57008
[18] W. P. Thurston, Hyperbolic structures on \(3\)-manifolds III: surface groups and \(3\)-manifolds with fiber over the circle , prépublication, 1980.
[19] W. P. Thurston, The topology and geometry of three-manifolds , 1977, notes de cours, Princeton Univ.
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