Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map

Abstract

In the paper [12], Yang conjectured that a nonelementary subgroup $G$ of $\mathrm{SL}(2, \mathbb{C})$ containing elliptic elements is discrete if for each elliptic element $g \in G$ the group $\langle f, g \rangle$ is discrete, where $f \in \mathrm{SL}(2,\mathbb{C})$ is a test map being loxodromic or elliptic. By embedding $\mathrm{SL}(2,\mathbb{C})$ into $\mathrm{U}(1,1; \mathbb{H})$, we give an affirmative answer to this question. As an application, we show that a nonelementary and nondiscrete subgroup of $\mathrm{Isom}(H^{3})$ must contain an elliptic element of order at least 3.