Full escape of mass for the diagonal group

TL;DR

This paper demonstrates the existence of infinite families of compact orbits of the diagonal group in the space of lattices that fully escape mass, revealing complex visitation patterns and introducing new tools for number field analysis.

Contribution

It proves the existence of full escape of mass for certain lattice orbits and develops novel methods for computing regulators of orders in number fields.

Findings

Existence of infinite families of compact orbits accumulating only on divergent orbits.

Sequences of orbits exhibit full escape of mass, sharpening previous partial results.

New tools for computing regulators of orders in number fields.

Abstract

Building on the work of Cassels we prove the existence of infinite families of compact orbits of the diagonal group in the space of lattices which accumulate only on the divergent orbit of the standard lattice. As a consequence, we prove the existence of full escape of mass for sequences of such orbits, sharpening known results about possible partial escape of mass. The topological complexity of the visits of these orbits to a given compact set in the space of lattices is analyzed. Along the way we develop new tools to compute regulators of orders in a number fields.