Primitive Curve Lengths on Pairs of Pants

TL;DR

This paper establishes bounds on the lengths of primitive curves on pairs of pants in discrete groups of PSL(2,C), using trace minimizing concepts and the Non-Euclidean Euclidean algorithm, advancing understanding of group discreteness.

Contribution

It introduces a new theorem linking trace minimizing to bounds on primitive curve lengths on pairs of pants in discrete groups.

Findings

Provides bounds on hyperbolic lengths of primitive curves

Connects trace minimizing to geometric properties of quotient surfaces

Utilizes the Non-Euclidean Euclidean algorithm for proofs

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.