Roots and rational powers in $$PSL(2,\mathbb {R})$$ P S L ( 2 , R ) and discreteness

Abstract

Let G be a finitely generated group of isometries of $$\mathbb {H}$$ , hyperbolic 2-space. The discreteness problem is to determine whether or not G is discrete. Even in the case of a two generator non-elementary subgroup of $$\mathbb {H}$$ (equivalently $$PSL(2,{\mathbb {R}})$$ ) the problem requires an algorithm (Gilman and Maskit in Mich Math J 38:13–32, 1991; Gilman in two-generator discrete subgroups of $$PSL(2, {\mathbb {R}})$$ , 1995). If G is discrete, one can ask when adjoining an nth root of a generator results in a discrete group. In this paper we first address the issue for pairs of hyperbolic generators in $$PSL(2, \mathbb {R})$$ with disjoint axes and obtain necessary and sufficient conditions for adjoining roots for the case when the two hyperbolics have a hyperbolic product and are what as known as stopping generators for the Gilman–Maskit algorithm (Gilman and Maskit 1991; Gilman and Keen in Cutting sequences and palindromes in geometry of Riemann surfaces, 2009). Stopping generators are generators that correspond to certain generalized Coxeter triangle group. We give an algorithmic solution in the other cases that are not stopping generators. Our solution applies to all other types of pairs of stopping generators that arise in what is known as the intertwining case. The results are geometrically motivated and stated as such, but also can be given computationally using the corresponding matrices.