All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups
- 14 Downloads
This is the second part of our works on the Hausdorff dimension of Schottky groups (Hou in Geom Topol 14:473–519, 2010). In this paper we prove that there exists a universal positive number \(\lambda >0\), such that up to a finite index, any finitely-generated non-elementary Kleinian group with a limit set of Hausdorff dimension \(<\lambda \) is a classical Schottky group. The proof relies on our previous works in Hou (Geom Topol 14:473–519, 2010; J Differ Geom 57:173–195, 2001), which provide the foundation for the general result of this paper. Our results can also be considered as a converse to the well-known theorem of Doyle (Acta Math 160:249–284, 1988) and Phillips–Sarnak (Acta Math 155:173–241, 1985).
The author would like to express appreciation to Benson Farb, Peter Sarnak for the opportunity to discuss this work in detail. The author also thanks Peter Shalen, Marc Culler for their many thoughts on this paper. The author is thankful to the referee for the many corrections and suggestions on the paper. This work is dedicated to my father: Shu Ying Hou.
- 1.Agol, I.: Tameness of hyperbolic \(3\)-manifolds. (2004) Arxiv arXiv:math/0405568
- 4.Bowen, L.: Cheeger constants and \(L^2\)-Betti numbers. Duke Math. J. 164(3), 569–615 (2013) (to appear) Google Scholar
- 5.Bridgeman, M.: Hausdorff dimension and Weil–Petersson extension to quasifuchsian space. 14, 2 (2010)Google Scholar
- 7.Calegari, D., Gabai, D.: Shrinkwrapping and the taming of hyperbolic \(3\)-manifolds. (2004) Arxiv arXiv:math/0407161
- 13.Gromov, M.: Hyperbolic Groups. In Essays in Group Theory. In: Gersten (ed.) M.S.R.I. Publ. 8. Springer, Berlin. pp 75-263 (1987)Google Scholar
- 16.Kapovich, M.: Homological dimension and critical exponent of Kleinian groups. GAFA 18, 2017–2054 (2009)Google Scholar
- 17.Kapovich, M.: Hyperbolic Manifolds and Discrete Groups: Lecture on Thurston’s Hyperbolization, Birkhauser’s series “Progress in mathematics” (2000)Google Scholar
- 18.Marden, A.: Schottky Groups and Circles. Contributions to Analysis (a Collection of Papers Dedicated to Lipman Bers), pp. 273–278. Academic Press, New York (1974)Google Scholar
- 21.Patterson, S.J.: Measures on Limit Sets of Kleinian Groups. Analytical and Geometrical Aspects of Hyperbolic Space, pp. 291–323. Cambrige University Press, Cambrige (1987)Google Scholar
- 22.Phillips, R., Sarnak, P.: The Laplacian for domains in hyperbolic space and limit sets of Kleinian groups. Acta Math. 155, 173–241 (1985)Google Scholar