Discreteness criterion in SL(2,$\bc$) by a test map

TL;DR

This paper proves that a non-elementary subgroup of SL(2,C) containing elliptic elements is discrete if all subgroups generated with a specific test map are discrete, confirming Yang's conjecture.

Contribution

It provides an affirmative proof of Yang's conjecture regarding discreteness criteria in SL(2,C) using a test map approach.

Findings

Confirmed Yang's conjecture on discreteness criteria

Established conditions for subgroup discreteness involving a test map

Enhanced understanding of subgroup structure in SL(2,C)

Abstract

In the paper (Osaka J. Math. {\bf 46}: 403-409, 2009), Yang conjectured that a non-elementary subgroup $G$ of $SL(2, \bc)$ containing elliptic elements is discrete if for each elliptic element $g\in G$ the group $< f, g >$ is discrete, where $f\in SL(2,\bc)$ is a test map which is loxodromic or elliptic. The purpose of this paper is to give an affirmative answer to this question.