Osaka Journal of Mathematics

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

Wensheng Cao

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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.

Article information

Osaka J. Math., Volume 49, Number 4 (2012), 901-907.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30F40: Kleinian groups [See also 20H10] 20H10: Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]


Cao, Wensheng. Discreteness criterion in $\mathrm{SL}(2,\mathbb{C})$ by a test map. Osaka J. Math. 49 (2012), no. 4, 901--907.

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