Discreteness Criteria and the Hyperbolic Geometry of Palindroms

Abstract

We consider non-elementary representations of two generator free groups in $PSL(2,\mathbb{C})$, not necessarily discrete or free, $G = < A, B >$. 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 {\sl 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 $\HH^3$ whether or not $G$ is discrete. We show that there is a {\sl 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 non-negative 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.