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Published online by Cambridge University Press: 04 March 2021
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 $G(3)$ .
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.