  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.
