Measure-scaling quasi-isometries

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