LOCAL COORDINATES FOR COMPLEX AND QUATERNIONIC HYPERBOLIC PAIRS

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...

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Published inJournal of the Australian Mathematical Society (2001) Vol. 113; no. 1; pp. 57 - 78
Main Authors GONGOPADHYAY, KRISHNENDU, KALANE, SAGAR B.
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.08.2022
Subjects
Classification
Conjugation
Orbits
Parameterization
Representations
51M10
loxodromic elements
surface group
character variety
quaternionic hyperbolic space
traces
37C15
15B57
20E45
complex hyperbolic space
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Summary: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 $G(3)$ .
Bibliography:ObjectType-Article-1
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ISSN:1446-7887
1446-8107
DOI:10.1017/S144678872100001X