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.
1
DER de Mathématiques, ENS Paris-Saclay, Gif-sur-Yvette, France
2
Institut für Mathematik und Informatik, Universität Greifswald, Germany
Blandine Galiay; Ines Kath. Lattices in the Four-Dimensional Hyperbolic Oscillator Group. Journal of Lie Theory, Tome 32 (2022) no. 4, pp. 1139-1170. http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a11/
@article{JOLT_2022_32_4_a11,
author = {Blandine Galiay and Ines Kath},
title = {Lattices in the {Four-Dimensional} {Hyperbolic} {Oscillator} {Group}},
journal = {Journal of Lie Theory},
pages = {1139--1170},
year = {2022},
volume = {32},
number = {4},
url = {http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a11/}
}
TY - JOUR
AU - Blandine Galiay
AU - Ines Kath
TI - Lattices in the Four-Dimensional Hyperbolic Oscillator Group
JO - Journal of Lie Theory
PY - 2022
SP - 1139
EP - 1170
VL - 32
IS - 4
UR - http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a11/
ID - JOLT_2022_32_4_a11
ER -
%0 Journal Article
%A Blandine Galiay
%A Ines Kath
%T Lattices in the Four-Dimensional Hyperbolic Oscillator Group
%J Journal of Lie Theory
%D 2022
%P 1139-1170
%V 32
%N 4
%U http://geodesic.mathdoc.fr/item/JOLT_2022_32_4_a11/
%F JOLT_2022_32_4_a11
