Quasi-isometries of rank one S-arithmetic lattices

TL;DR

This paper classifies the large-scale geometric structure of rank one S-arithmetic lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero, showing quasi-isometries are close to algebraic symmetries.

Contribution

It completes the quasi-isometric classification of these lattices by proving all quasi-isometries are near commensurators, extending previous results to a broader class.

Findings

Quasi-isometries are within finite distance of commensurators.

Classification of lattices up to quasi-isometry is achieved.

Extends known results to nondiscrete locally compact fields.

Abstract

We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator.