Primitive Curve Lengths on Pairs of Pants

Abstract

The problem of determining whether or not a non-elementary subgroup of $PSL(2,\CC)$ is discrete is a long standing one. The importance of two generator subgroups comes from J{\o}rgensen's inequality which has as a corollary the fact that a non-elementary subgroup of $PSL(2,\CC)$ is discrete if and only if every non-elementary two generator subgroup is. A solution even in the two-generator $PSL(2,\RR)$ case appears to require an algorithm that relies on a the concept of {\sl trace minimizing} that was initiated by Rosenberger and Purzitsky in the 1970's Their work has lead to many discreteness results and algorithms. Here we show how their concept of trace minimizing leads to a theorem that gives bounds on the hyperbolic lengths of curves on the quotient surface that are the images of primitive generators in the case where the group is discrete and the quotient is a pair of pants. The result follows as a consequence of the Non-Euclidean Euclidean algorithm.