TY - JOUR

T1 - Discreteness criteria and the hyperbolic geometry of palindromes

AU - Gilman, Jane

AU - Keen, Linda

PY - 2009/2/17

Y1 - 2009/2/17

N2 - We consider non-elementary representations of two generator free groups in PSL(2, C), not necessarily discrete or free, G =A, B. Aword 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 H3 whether or not G is discrete. We show that there is a core geodesic 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 nonnegative rationals and from the primitive elements to L; the second is that G is geometrically finite if and only if the axis of every non-parabolic palindromic word in G intersects 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.

AB - We consider non-elementary representations of two generator free groups in PSL(2, C), not necessarily discrete or free, G =A, B. Aword 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 H3 whether or not G is discrete. We show that there is a core geodesic 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 nonnegative rationals and from the primitive elements to L; the second is that G is geometrically finite if and only if the axis of every non-parabolic palindromic word in G intersects 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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U2 - https://doi.org/10.1090/S1088-4173-09-00191-X

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

M3 - Article

VL - 13

SP - 76

EP - 90

JO - Conformal Geometry and Dynamics

JF - Conformal Geometry and Dynamics

SN - 1088-4173

IS - 3

ER -