What question did this study set out to answer?

The aim is to parametrize and classify the lattices in the hyperbolic oscillator group.

March 29, 2026Open Access

Lattices in the Four-Dimensional Hyperbolic Oscillator Group

Key Points

The aim is to parametrize and classify the lattices in the hyperbolic oscillator group.
Developed a method for classification of lattices
Used automorphisms of the hyperbolic oscillator group
Investigated the structure of commensurability classes
Established a bijection between commensurability classes and real quadratic fields
Identified properties of lattices in the hyperbolic group
Demonstrated unique classification of lattices through group automorphisms

Abstract

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.

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