Measure-scaling quasi-isometries

TL;DR

This paper introduces measure-scaling quasi-isometries, explores their properties in different types of graphs, and computes the associated scaling groups, revealing invariance and structural differences among various groups.

Contribution

It defines measure-scaling quasi-isometries, characterizes their scaling groups in amenable graphs, and computes these groups for specific classes of groups, highlighting invariance properties.

Findings

Scaling groups are invariant under measure-scaling quasi-isometries.

All positive reals form the scaling group for lattices in Carnot, SOL, and Baumslag-Solitar groups.

Lamplighter groups have a scaling group equal to the positive rationals.

Abstract

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any $\kappa>0$, while for amenable ones there exists at most one possible such $\kappa$. For an amenable graph $X$, we show that the set of possible $\kappa$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a given locally compact group have same scaling groups. We compute the scaling group in a number of cases. For instance it is all of $\mathbb{R}_{>0}$ for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup $\mathbb{Q}_{>0}$ for lamplighter groups over finitely presented amenable groups.