Density of shapes of periodic tori in the cubic case

TL;DR

This paper proves that the shapes of all periodic tori in a specific mathematical setting are dense in a certain space, implying the density of shapes of unit groups of totally real cubic orders.

Contribution

It establishes the density of shapes of periodic tori in the space of lattices, connecting to the density of unit group shapes of totally real cubic orders.

Findings

Shapes of all periodic tori are dense in the shape space.

Implication for density of unit groups of totally real cubic orders.

Provides a link between dynamical systems and algebraic number theory.

Abstract

Consider the compact orbits of the $\mathbb{R}^2$ action of the diagonal group on $\operatorname{SL}(3,\mathbb{R})/\operatorname{SL}(3,\mathbb{Z})$, the so-called periodic tori. For any periodic torus, the set of periods of the orbit forms a lattice in $\mathbb{R}^2$. Such a lattice, re-scaled to covolume one, gives a shape point in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. We prove that the shapes of all periodic tori are dense in $\operatorname{SL}(2,\mathbb{R})/\operatorname{SL}(2,\mathbb{Z})$. This implies the density of shapes of the unit groups of totally real cubic orders.