Approximations of the Riley slice

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