Lattices in the four-dimensional split oscillator group

TL;DR

This paper classifies lattices in the four-dimensional split oscillator group, revealing their commensurability classes correspond to real quadratic fields, thus advancing understanding of its geometric and algebraic structure.

Contribution

It provides a parametrization and classification method for lattices in the split oscillator group, linking their classes to real quadratic fields.

Findings

Lattices in the split oscillator group are parametrized.

Commensurability classes correspond to real quadratic fields.

A classification method via automorphisms is developed.

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 split oscillator group (sometimes also hyperbolic 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.