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Cartan decomposition for $\mathrm {Sp}(1,n)$ and some applications to group dynamics

Part of: Basic linear algebra Special matrices

Published online by Cambridge University Press:  28 October 2025

Angel Cano  and
Hector Abrahan Castro Morales
Angel Cano*
Affiliation:
UCIM, Universidad Nacional Autónoma de México, Mexico e-mail: hector.castro@im.unam.mx
Hector Abrahan Castro Morales
Affiliation:
UCIM, Universidad Nacional Autónoma de México, Mexico e-mail: hector.castro@im.unam.mx
*
e-mail: angelcano@im.unam.mx
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Abstract

In this article, we present an elementary proof of the Cartan decomposition theorem for the group $ \mathrm {Sp}(1, n) $. As an application, we determine the largest regular domain for discrete quaternionic hyperbolic groups acting on $ \mathbb {HP}^n $. Furthermore, we demonstrate that Bers’ simultaneous uniformization and Köebe’s retrosection theorem fail to hold for higher-dimensional quaternionic Kleinian groups.

Keywords

Quaternion linear algebraCartan decomposition applicationsKleinian groups

MSC classification

Primary: 15B33: Matrices over special rings (quaternions, finite fields, etc.) 15A23: Factorization of matrices 15B57: Hermitian, skew-Hermitian, and related matrices

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 17
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The research of A.C. was partially supported by SECIHTI-SNI 104023 and PAPIIT-UNAM IN112424.

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