abstract. the problem of determining whether or not a nonelementary subgroup of p sl(2, c) is discrete is a long standing
one. the importance of two generator subgroups comes from
jørgensen’s inequality which has as a corollary the fact that a nonelementary subgroup of p sl(2, c) is discrete if and only if every
non-elementary two generator subgroup is. a solution even in the
two-generator p sl(2, r) case appears to require an algorithm that
relies on a the concept of 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.