Torsion free groups generated by a pair of rational parabolic Möbius transformations

Abstract

Let T be a subgroup of PSL (2, ℚ) generated by a pair of rational parabolic matrices P1, P2, and let ℐ be the Jøgensen number. We prove that T has a non-trivial element of finite order if and only if ℐ = 4/n2 or ℐ = 9/n2 for some non-zero integer n.