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 $PSL(2,\mathbb {C})$, 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.

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Additional 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

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.