Geodesic
Lattices in the Four-Dimensional Hyperbolic Oscillator Group
Journal of Lie theory, Tome 32 (2022) no. 4, pp. 1139-117
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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.
Classification : 53C50, 22E40, 57S30
Mots-clés : Solvable Lie group, lattice, biinvariant pseudo-Riemannian metric 
@article{JLT_2022_32_4_JLT_2022_32_4_a11,
     author = {B. Galiay and I. Kath},
     title = {Lattices in the {Four-Dimensional} {Hyperbolic} {Oscillator} {Group}},
     journal = {Journal of Lie theory},
     pages = {1139--117},
     year = {2022},
     volume = {32},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JLT_2022_32_4_JLT_2022_32_4_a11/}
}
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B. Galiay; I. Kath. Lattices in the Four-Dimensional Hyperbolic Oscillator Group. Journal of Lie theory, Tome 32 (2022) no. 4, pp. 1139-117. http://geodesic.mathdoc.fr/item/JLT_2022_32_4_JLT_2022_32_4_a11/