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Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions

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Abstract

Using the rings of Lipschitz and Hurwitz integers \(\mathbb {H}(\mathbb {Z})\) and \(\mathbb {H}ur(\mathbb {Z})\) in the quaternion division algebra \(\mathbb {H}\), we define several Kleinian discrete subgroups of \(PSL(2,\mathbb {H})\). We define first a Kleinian subgroup \(PSL(2,\mathfrak {L})\) of \(PSL(2,\mathbb {H}(\mathbb {Z}))\). This group is a generalization of the modular group \(PSL(2,\mathbb {Z})\). Next we define a discrete subgroup \(PSL(2,\mathfrak {H})\) of \(PSL(2,\mathbb {H})\) which is obtained by using Hurwitz integers. It contains as a subgroup \(PSL(2,\mathfrak {L})\). In analogy with the classical modular case, these groups act properly and discontinuously on the hyperbolic quaternionic half space. We exhibit fundamental domains of the actions of these groups and determine the isotropy groups of the fixed points and describe the orbifold quotients \(\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {L})\) and \(\mathbf {H}_{\mathbb {H}}^1/PSL(2,\mathfrak {H})\) which are quaternionic versions of the classical modular orbifold and they are of finite volume. Finally we give a thorough study of their descriptions by Lorentz transformations in the Lorentz–Minkowski model of hyperbolic 4-space.

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Notes

  1. By this we mean that a \(2\times 2\) quaternionic matrix A has a right and left inverse; in [4] it is shown that this is equivalent for A to have non zero Dieudonné determinant (see [3]).

  2. In the following sense; T sends every point of a hyperbolic geodesic parametrized by arc length \(\gamma (s)\), passing through 1 at time 0 (i.e. such that \(\gamma (0)=1\)), to its opposite \(\gamma (-s)\).

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Acknowledgements

This research paper has been made possible thanks to the financial support generously given by FORDECYT 265667 (Mexico) and the Italian institute Gruppo Nazionale Strutture Algebriche, Geometriche e Applicazione (GNSAGA) of the Istituto Nazionale di Alta Matematica (INdAM) “F. Severi”. Finally special thanks go to the International Centre for Theoretical Physics (ICTP) “A. Salam” of Trieste. The first and second named authors were financed by grant IN106817, PAPIIT, DGAPA, Universidad Nacional Autónoma de México and CONACyT (Mexico). The last author was also partially supported by Ministero Istruzione Università e Ricerca MIUR Progetto di Ricerca di Interesse Nazionale PRIN “Proprietà Geometriche delle Varietà Reali e Complesse”

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Correspondence to Alberto Verjovsky.

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Dedicated to Bill Goldman on occasion of his 60th birthday.

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Díaz, J.P., Verjovsky, A. & Vlacci, F. Quaternionic Kleinian modular groups and arithmetic hyperbolic orbifolds over the quaternions. Geom Dedicata 192, 127–155 (2018). https://doi.org/10.1007/s10711-017-0288-z

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