2021/05/11 by Anthony Genevois, Romain Tessera, Genevois, Anthony +1 · 1 citation
Mathematics · #20F65 #20F69 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #Group Theory (math.GR) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2105.04883
openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.