The combinatorics of Farey words and their traces

TL;DR

This paper introduces a new family of Farey polynomials with applications in 3-manifold geometry, providing recursive combinatorial definitions, examples, and conjectures relevant to algebraic combinatorics and hypergeometric functions.

Contribution

It develops a recursive combinatorial framework for Farey polynomials, extending known cases and connecting them to geometric and topological properties of 3-manifolds.

Findings

New family of 3-variable Farey polynomials introduced

Recursive combinatorial definitions extended beyond known cases

Applications to classifying rank-two subgroups of PSL(2,C) and group discreteness

Abstract

We introduce a family of 3-variable "Farey polynomials" that are closely connected with the geometry and topology of $3$-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-complex-dimensional deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank-two subgroups of PSL(2,C) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.