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.