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ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 101 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 16 August 2019, pp. 283-293
- Print publication:
- April 2020
-
- Article
- Export citation
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Let
$\mathbf{H}_{\mathbb{H}}^{n}$ denote the 
$n$-dimensional quaternionic hyperbolic space. The linear group 
$\text{Sp}(n,1)$ acts on 
$\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup 
$G$ of 
$\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on 
$\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of 
$\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup 
$G$ of 
$\text{Sp}(n,1)$ is discrete if for every loxodromic element 
$g\in G$ the two-generator subgroup 
$\langle f,gfg^{-1}\rangle$ is discrete, where the generator 
$f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from 
$G$.
