Divergence of separated nets with respect to displacement equivalence

TL;DR

This paper introduces a hierarchy of equivalence relations on separated nets in Euclidean space based on displacement functions, revealing a spectrum from bounded displacement to complete equivalence, and compares these with existing notions like bilipschitz equivalence.

Contribution

It defines a new family of $oldsymbol{ extit{ extbf{ extphi}}}$-displacement equivalences, analyzing their properties and relationships with known equivalence notions in geometric group theory.

Findings

Spectrum of $ extit{ extphi}$-displacement equivalences ranges from bounded to indiscrete.

Different $ extit{ extphi}$ functions induce distinct asymptotic classes of equivalence.

Comparison with bilipschitz equivalence highlights differences and similarities.

Abstract

We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions $\phi\colon (0,\infty)\to(0,\infty)$. Two separated nets are called $\phi$-displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii $R$, displaces points of norm at most $R$ by something of order at most $\phi(R)$. We show that the spectrum of $\phi$-displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded $\phi$, to the indiscrete equivalence relation, coresponding to $\phi(R)\in \Omega(R)$, in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of $\phi$-displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of $\phi(R)$ for $R\to\infty$. We further undertake a comparison of our notion of $\phi$-displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of $\phi$-displacement equivalence with that of bilipschitz equivalence.