Algebraic convergence of Schottky groups

Author:
Richard D. Canary

Journal:
Trans. Amer. Math. Soc. **337** (1993), 235-258

MSC:
Primary 30F40; Secondary 30F60, 32G15, 57M07, 57S30

DOI:
https://doi.org/10.1090/S0002-9947-1993-1137257-9

MathSciNet review:
1137257

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A discrete faithful representation of the free group on generators into is said to be a Schottky group if is homeomorphic to a handlebody (where is the domain of discontinuity for 's action on the sphere at infinity for ). Schottky space , the space of all Schottky groups, is parameterized by the quotient of the Teichmüller space of the closed surface of genus by where is the group of (isotopy classes of) homeomorphisms of which extend to homeomorphisms of which are homotopic to the identity. Masur exhibited a domain of discontinuity for 's action on (the space of projective measured laminations on ), so may be appended to as a boundary. Thurston conjectured that if a sequence of Schottky groups converged into , then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston's conjecture in the case where is homeomorphic to and the length in of the closed geodesic(s) in the homotopy class of the boundary of is bounded above by some constant .

**[1]**W. Abikoff,*The real analytic theory of Teichmüller space*, Lecture Notes in Math., vol. 820, Springer-Verlag, 1980. MR**590044 (82a:32028)****[2]**A. F. Beardon,*The geometry of discrete groups*, Graduate Texts in Math., vol. 91, Springer-Verlag, 1983. MR**698777 (85d:22026)****[3]**L. Bers,*Spaces of kleinian groups*, Maryland Conference in Several Complex Variables. I, Lectures Notes in Math., vol. 155, Springer-Verlag, 1970, pp. 9-34. MR**0271333 (42:6216)****[4]**L. Bers and F. Gardiner,*Fricke spaces*, Adv. in Math.**62**(1986), 249-284. MR**866161 (88g:32038)****[5]**M. Bestvina,*Degenerations of hyperbolic space*, Duke Math. J.**56**(1988), 143-161. MR**932860 (89m:57011)****[6]**F. Bonahon,*Bouts des variétés hyperboliques de dimension*, prepublicationes de Orsay, 1985.**[7]**-,*Bouts des variétés hyperboliques de dimension*, Ann. of Math.**124**(1986), pp. 71-158. MR**847953 (88c:57013)****[8]**-,*The geometry of Teichmüller space via geodesic currents*, Invent. Math.**92**(1988), 139-162. MR**931208 (90a:32025)****[9]**-,*Geodesic currents on hyperbolic groups*, Arboreal Group Theory, edited by R. C. Alperin, Springer-Verlag, New York, 1991, pp. 143-168.**[10]**R. D. Canary,*The Poincaré metric and a conformal version of a theorem of Thurston*, Duke Math. J.**64**(1991), 349-359. MR**1136380 (92k:57020)****[11]**-,*Hyperbolic structures on*-*manifolds with compressible boundary*, Ph.D. thesis, Princeton Univ., 1989.**[12]**R. D. Canary, D. B. A. Epstein, and P. Green,*Notes on notes of Thurston*, Analytical and Geometrical Aspects of Hyperbolic Spaces, Cambridge Univ. Press, 1987, pp. 3-92. MR**903850 (89e:57008)****[13]**A. Fathi, F. Laudenbach and V. Poenaru,*Travaux de Thurston sur les surfaces*, Asterique**66-67**(1979). MR**568308 (82m:57003)****[14]**W. J. Floyd,*Group completions and limit sets of Kleinian groups*, Invent. Math.**57**(1980), 205-218. MR**568933 (81e:57002)****[15]**-,*Hyperbolic manifolds obtained by adding a*-*handle to a*-*manifold with compressible boundary*, preprint.**[16]**C. McA. Gordon,*On primitive sets of loops in the boundary of the handlebody*, Topology Appl.**27**(1987), 285-299. MR**918538 (88k:57013)****[17]**J. Hempel, -*manifolds*, Ann. of Math. Studies, no. 86, Princeton Univ. Press, 1976.**[18]**W. Jaco,*Adding a*-*handle to a*-*manifold*:*an application to property*, Proc. Amer. Math. Soc.**92**(1984), 282-292. MR**754723 (86b:57006)****[19]**T. Jorgensen,*On discrete groups of Möbius transformations*, Amer. J. Math.**98**(1976), 739-749. MR**0427627 (55:658)****[20]**T. Jorgensen and A. Marden,*Algebraic and geometric convergence of Kleinian groups*, Math. Scand.**66**(1990), 47-72. MR**1060898 (91f:30068)****[21]**S. P. Kerckhoff,*The Nielsen realization problem*, Ann. of Math.**117**(1983), 235-265. MR**690845 (85e:32029)****[22]**-,*The measure of the limit set of the handlebody group*, Topology**29**(1990), 27-40. MR**1046623 (91b:57011)****[23]**I. Kra,*On spaces of Kleinian groups*, Comment. Math. Helv.**47**(1972), 53-69. MR**0306485 (46:5611)****[24]**B. Maskit,*Self-maps of Kleinian groups*, Amer. J. Math.**93**(1971), 840-856. MR**0291453 (45:544)****[25]**-,*Kleinian groups*, Springer-Verlag, 1988. MR**959135 (90a:30132)****[26]**H. Masur,*Measured foliations and handlebodies*, Ergodic Theory Dynamical Systems**6**(1986), 99-116. MR**837978 (87i:57011)****[27]**J. W. Morgan,*On Thurston's uniformization theorem for three-dimensional manifolds*, The Smith Conjecture, edited by J. Morgan and H. Bass, Academic Press, 1984, pp. 37-125. MR**758464****[28]**J. W. Morgan and J. P. Otal,*Relative growth rates of closed geodesics on a surface under varying hyperbolic structures*, preprint. MR**1214228 (94d:57005)****[29]**J. W. Morgan and P. Shalen,*Valuations, trees and degenerations of hyperbolic structures*. I, Ann. of Math.**120**(1984), 401-476. MR**769158 (86f:57011)****[30]**-,*Degeneration of hyperbolic structures*II:*Measured laminations in*-*manifolds*, Ann. of Math.**127**(1988), 403-456. MR**932305 (89e:57010a)****[31]**-,*Degeneration of hyperbolic structures*III:*Actions of*-*manifold groups on trees and Thurston's compactification theorem*, Ann. of Math.**127**(1988), 457-519. MR**942518 (89e:57010b)****[32]**K. Ohshika,*On limits of quasi-conformal deformations of Kleinian groups*, Math. Z.**201**(1989), 167-176. MR**997219 (90f:30058)****[33]**J. P. Otal,*Courants geodesiques et produits libres*, preprint.**[34]**F. Paulin,*Topologie de Gromov équivariante, structures hyperboliques et arbres réels*, Invent. Math.**94**(1988), 53-80. MR**958589 (90d:57015)****[35]**R. Skora,*Geometric actions of surface groups on*-*trees*, Comment. Math. Helv.**65**(1990), 519-533. MR**1078095 (91k:20035)****[36]**W. Thurston,*Hyperbolic structures on*-*manifolds*, I:*Deformation of acylindrical manifolds*, Ann. of Math.**124**(1986), 203-246. MR**855294 (88g:57014)****[37]**-,*Hyperbolic structures on*-*manifolds*, II:*Surface groups and*-*manifolds which fiber over the circle*, preprint.**[38]**-,*Hyperbolic structures on*-*manifold*, III:*Deformations of*-*manifolds with incompressible boundary*, preprint.**[39]**-,*The geometry and topology of*-*manifolds*, lecture notes.**[40]**-,*Minimal stretch maps between hyperbolic surfaces*, preprint.**[41]**M. Wolf,*The Teichmüller theory of harmonic maps*, J. Defferential Geometry**29**(1989), 449-479. MR**982185 (90h:58023)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30F40,
30F60,
32G15,
57M07,
57S30

Retrieve articles in all journals with MSC: 30F40, 30F60, 32G15, 57M07, 57S30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1137257-9

Article copyright:
© Copyright 1993
American Mathematical Society