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ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES
Published online by Cambridge University Press: 16 August 2019
Abstract
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 $G$.
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- Research Article
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- Copyright
- © 2019 Australian Mathematical Publishing Association Inc.
Footnotes
The first author acknowledges partial support from SERB MATRICS grant MTR/2017/000355; the third author is supported by NBHM-SRF.