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LOCAL COORDINATES FOR COMPLEX AND QUATERNIONIC HYPERBOLIC PAIRS

Part of: Structure and classification of infinite or finite groups Special matrices Real and complex geometry Smooth dynamical systems: general theory

Published online by Cambridge University Press:  04 March 2021

KRISHNENDU GONGOPADHYAY Open the ORCID record for KRISHNENDU GONGOPADHYAY [Opens in a new window]  and
SAGAR B. KALANE
KRISHNENDU GONGOPADHYAY*
Affiliation:
Indian Institute of Science Education and Research (IISER) Mohali, Knowledge City, Sector 81, S.A.S. Nagar 140306, Punjab, India e-mail: krishnendu@iisermohali.ac.in
SAGAR B. KALANE
Affiliation:
Indian Institute of Science Education and Research (IISER) Pune, Dr. Homi Bhabha Road, Pashan, Pune 411008, India e-mail: sagark327@gmail.com
*
e-mail: krishnendug@gmail.com
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Abstract

Let $G(n)={\textrm {Sp}}(n,1)$ or ${\textrm {SU}}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called ‘nonsingular’, were introduced by Gongopadhyay and Parsad for ${\textrm {SU}}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dimension. For $n=3$, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed (genus $g \geq 2$) oriented surface into $G(3)$.

Keywords

character varietycomplex hyperbolic spaceloxodromic elementsquaternionic hyperbolic spacesurface grouptraces

MSC classification

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification
Secondary: 51M10: Hyperbolic and elliptic geometries (general) and generalizations 15B57: Hermitian, skew-Hermitian, and related matrices 20E45: Conjugacy classes

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 1 , August 2022 , pp. 57 - 78
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Ben Martin

K. Gongopadhyay acknowledges partial support from SERB-DST MATRICS project MTR/2017/000355. S. B. Kalane is supported by an IISER Pune Institute post doctoral fellowship.

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