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ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES

Published online by Cambridge University Press:  16 August 2019

KRISHNENDU GONGOPADHYAY
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab 140306, India email krishnendu@iisermohali.ac.in, krishnendug@gmail.com
MUKUND MADHAV MISHRA
Affiliation:
Department of Mathematics, Hansraj College, University of Delhi, Delhi110007, India email mukund.math@gmail.com
DEVENDRA TIWARI*
Affiliation:
Department of Mathematics, University of Delhi, Delhi 110007, India email devendra9.dev@gmail.com

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 $G$.

Type
Research Article
Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

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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