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

DOI:
https://doi.org/10.1090/S1088-4173-09-00191-X

Published electronically:
February 17, 2009

MathSciNet review:
2476657

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider non-elementary representations of two generator free groups in , not necessarily discrete or free, . A word in and , , 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 in whether or not is discrete. We show that there is a *core geodesic* in the convex hull of the limit set of 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 ; the second is that is geometrically finite if and only if the axis of every non-parabolic palindromic word in intersects 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.

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Additional Information

**Jane Gilman**

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

Email:
gilman@rutgers.edu

**Linda Keen**

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

Email:
Linda.keen@lehman.cuny.edu

DOI:
https://doi.org/10.1090/S1088-4173-09-00191-X

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.