# Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

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## Discreteness criteria and the hyperbolic geometry of palindromesHTML 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

## 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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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