Measure-scaling quasi-isometries

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.

19 pages, 1 figure. Comments are welcome