Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-K-to-one in a natural sense for some kappa > 0. For non-amenable graphs, all quasi-isometrics 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 R->0 that we call the (measure-)scaling group of X. This group is invariant under measure-scaling quasi-isometrics. 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 R->(0) for lattices in Carrot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup Q(>0)( )for lamplighter groups over finitely presented amenable groups.