## Discreteness criteria and the hyperbolic geometry of palindromes

HTML articles powered by AMS MathViewer

- by Jane Gilman and Linda Keen
- Conform. Geom. Dyn.
**13**(2009), 76-90 - DOI: https://doi.org/10.1090/S1088-4173-09-00191-X
- Published electronically: February 17, 2009
- PDF | Request permission

## 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

- Alan F. Beardon,
*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777**, DOI 10.1007/978-1-4612-1146-4 - 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. MR**903850** - 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** - 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**, DOI 10.1515/9783110849455 - Jane Gilman and Linda Keen,
*The geometry of two generator groups: hyperelliptic handlebodies*, Geom. Dedicata**110**(2005), 159–190. MR**2136025**, DOI 10.1007/s10711-004-6556-8 - 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**, DOI 10.1090/conm/311/05455 - Gilman, Jane and Keen, Linda,
*Enumerating Plaindromes in Rank Two Free Groups*, submitted. - Michael Kapovich,
*Hyperbolic manifolds and discrete groups*, Progress in Mathematics, vol. 183, Birkhäuser Boston, Inc., Boston, MA, 2001. MR**1792613** - 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**, DOI 10.1017/CBO9780511618789 - 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**, DOI 10.1016/0040-9383(93)90048-Z - Magnus, Wilhelm; Karass, Abraham; and Solitar, Donald,
*Combinatorial Group Theory*(1966) John Wiley & Sons, NYC. - 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** - Thurston, William P.,
*The Geometry and Topology of Three Manifolds*, lecture notes, Princeton Univ., Princeton, N.J. (1979).

## Bibliographic Information

**Jane Gilman**- Affiliation: Department of Mathematics, Rutgers University, Newark, New Jersey 07079
- MR Author ID: 190609
- Email: gilman@rutgers.edu
**Linda Keen**- Affiliation: Department of Mathematics, Lehman College and Graduate Center, CUNY, Bronx, New York, New York 10468
- MR Author ID: 99725
- Email: Linda.keen@lehman.cuny.edu
- 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 - © Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Conform. Geom. Dyn.
**13**(2009), 76-90 - MSC (2000): Primary 30F10, 30F35, 30F40; Secondary 14H30, 22E40
- DOI: https://doi.org/10.1090/S1088-4173-09-00191-X
- MathSciNet review: 2476657