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Journal of Lie Theory 32 (2022), No. 4, 1139--1170
Copyright Heldermann Verlag 2022

Lattices in the Four-Dimensional Hyperbolic Oscillator Group

Blandine Galiay
DER de Mathématiques, ENS Paris-Saclay, Gif-sur-Yvette, France

Ines Kath
Institut für Mathematik und Informatik, Universität Greifswald, Germany

Besides the oscillator group, there is another four-dimensional non-abelian solvable Lie group that admits a bi-invariant pseudo-Riemannian metric. It is called hyperbolic oscillator group (sometimes also split oscillator group or Boidol's group). We parametrise the set of lattices in this group and develop a method to classify these lattices up automorphisms of the ambient group. We show that their commensurability classes are in bijection with the set of real quadratic fields.

Keywords: Solvable Lie group, lattice, biinvariant pseudo-Riemannian metric.

MSC: 53C50, 22E40, 57S30.

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