Separated Nets in Nilpotent Groups

TL;DR

This paper extends the study of separated nets from Euclidean spaces to nilpotent Lie groups, demonstrating the existence of non-biLipschitz equivalent nets and exploring their properties through generalized constructions and criteria.

Contribution

It introduces a framework for separated nets in nilpotent groups, including new constructions and criteria, expanding understanding beyond Euclidean spaces.

Findings

Existence of non-biLipschitz equivalent separated nets in nilpotent groups

Construction of separated nets via generalized cut-and-project methods

Identification of exotic perturbations using a generalized Laczkovich criterion

Abstract

In this paper we generalize several results on separated nets in Euclidean space to separated nets in connected simply connected nilpotent Lie groups. We show that every such group $G$ contains separated nets that are not biLipschitz equivalent. We define a class of separated nets in these groups arising from a generalization of the cut-and-project quasi-crystal construction and show that generically any such separated net is bounded displacement equivalent to a separated net of constant covolume. In addition, we use a generalization of the Laczkovich criterion to provide `exotic' perturbations of such separated nets.