Measure-scaling quasi-isometries

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-κ-to-one in a natural sense for some κ> 0. For non-amenable graphs, all quasi-isometries are quasi-κ-to-one for any κ> 0 , while for amenable ones there exists at most one possible such κ. For an amenable graph X, we show that the set of possible κ forms a subgroup of 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 R_{> 0} for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup Q_{> 0} for lamplighter groups over finitely presented amenable groups.