The combinatorics of Farey words and their traces

Abstract

The set of Kleinian groups which are free on two parabolic generators is parameterised by the closed Riley slice of Schottky space. A 'Farey word' is a word in such a group which represents a non-boundary-parallel geodesic that can be pinched down to a puncture; in the interior of the Riley slice such a word is loxodromic, and the pinching process corresponds to deforming the word to be parabolic. Keen and Series showed that the geometry of the Riley slice is detected by the real loci of the trace polynomials of these words. We study these trace polynomials from a combinatorial viewpoint, and give a recursion formula for them which enables efficient calculation of the polynomials without performing matrix multiplication; we also present some intriguing examples to show that there is much still to be learned about them.