Remote Access Conformal Geometry and Dynamics
Green Open Access

Conformal Geometry and Dynamics

ISSN 1088-4173



Discreteness criteria and the hyperbolic geometry of palindromes

Authors: Jane Gilman and Linda Keen
Journal: Conform. Geom. Dyn. 13 (2009), 76-90
MSC (2000): Primary 30F10, 30F35, 30F40; Secondary 14H30, 22E40
Published electronically: February 17, 2009
MathSciNet review: 2476657
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider non-elementary representations of two generator free groups in $ PSL(2,\mathbb{C})$, not necessarily discrete or free, $ G = \langle A, B \rangle$. A word in $ A$ and $ B$, $ W(A,B)$, is a palindrome if it reads the same forwards and backwards. A word in a free group is primitive if it is part of a minimal generating set. Primitive elements of the free group on two generators can be identified with the positive rational numbers. We study the geometry of palindromes and the action of $ G$ in $ {\mathbb{H}}^3$ whether or not $ G$ is discrete. We show that there is a core geodesic $ {\mathbf{L}}$ in the convex hull of the limit set of $ G$ and use it to prove three results: the first is that there are well-defined maps from the non-negative rationals and from the primitive elements to $ {\mathbf{L}}$; the second is that $ G$ is geometrically finite if and only if the axis of every non-parabolic palindromic word in $ G$ intersects $ {\mathbf{L}}$ in a compact interval; the third is a description of the relation of the pleating locus of the convex hull boundary to the core geodesic and to palindromic elements.

References [Enhancements On Off] (What's this?)

  • 1. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777
  • 2. 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. MR 903850
  • 3. D. B. A. Epstein and A. Marden, Convex hulls in hyperbolic space, a theorem of Sullivan, and measured pleated surfaces, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 113–253. MR 903852
  • 4. Werner Fenchel, Elementary geometry in hyperbolic space, de Gruyter Studies in Mathematics, vol. 11, Walter de Gruyter & Co., Berlin, 1989. With an editorial by Heinz Bauer. MR 1004006
  • 5. Jane Gilman and Linda Keen, The geometry of two generator groups: hyperelliptic handlebodies, Geom. Dedicata 110 (2005), 159–190. MR 2136025, 10.1007/s10711-004-6556-8
  • 6. Jane Gilman and Linda Keen, Word sequences and intersection numbers, Complex manifolds and hyperbolic geometry (Guanajuato, 2001) Contemp. Math., vol. 311, Amer. Math. Soc., Providence, RI, 2002, pp. 231–249. MR 1940172, 10.1090/conm/311/05455
  • 7. Gilman, Jane and Keen, Linda, Enumerating Plaindromes in Rank Two Free Groups, submitted.
  • 8. Michael Kapovich, Hyperbolic manifolds and discrete groups, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1792613
  • 9. Linda Keen and Nikola Lakic, Hyperbolic geometry from a local viewpoint, London Mathematical Society Student Texts, vol. 68, Cambridge University Press, Cambridge, 2007. MR 2354879
  • 10. Linda Keen and Caroline Series, Pleating coordinates for the Maskit embedding of the Teichmüller space of punctured tori, Topology 32 (1993), no. 4, 719–749. MR 1241870, 10.1016/0040-9383(93)90048-Z
  • 11. Magnus, Wilhelm; Karass, Abraham; and Solitar, Donald, Combinatorial Group Theory (1966) John Wiley & Sons, NYC.
  • 12. D. A. Každan and G. A. Margulis, A proof of Selberg’s hypothesis, Mat. Sb. (N.S.) 75 (117) (1968), 163–168 (Russian). MR 0223487
  • 13. Thurston, William P., The Geometry and Topology of Three Manifolds, lecture notes, Princeton Univ., Princeton, N.J. (1979).

Similar Articles

Retrieve articles in Conformal Geometry and Dynamics of the American Mathematical Society with MSC (2000): 30F10, 30F35, 30F40, 14H30, 22E40

Retrieve articles in all journals with MSC (2000): 30F10, 30F35, 30F40, 14H30, 22E40

Additional Information

Jane Gilman
Affiliation: Department of Mathematics, Rutgers University, Newark, New Jersey 07079

Linda Keen
Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York, New York 10468

Received by editor(s): December 29, 2008
Published electronically: February 17, 2009
Additional Notes: The first author was supported in part by the Rutgers Research Council and Yale University
The second author was supported in part by the PSC-CUNY
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.