Rational powers of generators of Möbius groups (Q1199274)

Let \(a,b\in PSL(2,\mathbb{R})\). The best technique available to decide if the group \(\langle a,b\rangle\) generated by \(a\) and \(b\) is discrete or not is the algorithm described by \textit{N. Purzitsky} [Math. Z. 126, 209-223 (1972; Zbl 0221.20064) and ibid. 147, 87-92 (1976; Zbl 0305.20025)] and \textit{G. Rosenberger} [Math. Ann. 199, 213-227 (1972; Zbl 0244.20057) and Arch. Math. 46, 198-204 (1986; Zbl 0563.20043)]. An interesting additional problem is the following. Describe algorithmically conditions for the discreteness of \(\langle a,b\rangle\) if \(\langle a^ \alpha,b^ \beta\rangle\) is discrete for some \(\alpha,\beta \in \mathbb{Z} \setminus \{0\}\). Such conditions are given for \(a\) and \(b\) elliptic or parabolic.