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Published online by Cambridge University Press:  16 August 2019

Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab 140306, India email,
Department of Mathematics, Hansraj College, University of Delhi, Delhi110007, India email
Department of Mathematics, University of Delhi, Delhi 110007, India email


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 $G$.

Research Article
© 2019 Australian Mathematical Publishing Association Inc.

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The first author acknowledges partial support from SERB MATRICS grant MTR/2017/000355; the third author is supported by NBHM-SRF.


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