On generalized J\o{}rgensen inequality in ${\rm SL}(2, \mathbb C)$

TL;DR

This paper proves that a previously established generalized Jørgensen inequality for two-generator subgroups of SL(2,C), with one loxodromic generator, is actually strict, strengthening the original result.

Contribution

The paper demonstrates that the existing generalized Jørgensen inequality in SL(2,C) is strict, providing a sharper bound than previously known.

Findings

The inequality is strict, not just non-strict.

Strengthens the understanding of subgroup discreteness criteria.

Provides a more precise condition for two-generator subgroups in SL(2,C).

Abstract

Wang, Jiang and Cao have obtained a generalized version of the J\o{}rgensen inequality in Proc. Indian Acad. Sci. Math. Sci., 123(2):245--251, 2013, for two generator subgroups of ${\rm SL}(2, \mathbb C)$ where one of the generators is loxodromic. We prove that their inequality is strict.