Approximations of the Riley slice

TL;DR

This paper develops a method to identify points within the Riley slice of Schottky groups using geometric and dynamical systems techniques, advancing understanding of its structure and related groups.

Contribution

It introduces explicit neighborhoods and a provable method for determining Riley slice membership, linking geometric, algebraic, and dynamical perspectives.

Findings

Explicit half-space neighborhoods in the Riley slice.

A dynamical systems interpretation of the Riley slice.

A framework for identifying arithmetic generalized triangle groups.

Abstract

Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify `half-space' neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.