Closed orbits for the diagonal group and well-rounded lattices

TL;DR

This paper extends McMullen's result by proving that all closed orbits of the diagonal group action on the space of lattices contain well-rounded lattices, broadening the understanding of orbit structure.

Contribution

The paper generalizes previous work by showing that every closed orbit, not just compact ones, contains a well-rounded lattice.

Findings

All closed orbits contain well-rounded lattices.

Extension of McMullen's result to all closed orbits.

Enhanced understanding of lattice orbit structures.

Abstract

Curt McMullen showed that every compact orbit for the action of the diagonal group on the space of lattices contains a well-rounded lattice. We extend this to all closed orbits.