2025/02/03 by Vincent Dumoncel, Dumoncel, Vincent
Engineering · Mathematics · #20F65 #20F69 #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Group Theory (math.GR) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2502.01849
openalex publication_date 2025/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A quasi-isometry between two connected graphs is measure-scaling if one can control precisely the sizes of pre-images of finite subsets. Such a notion is motivated by the work of Eskin-Fisher-Whyte on lamplighters over ℤ and the work of Dymarz on biLipschitz equivalences of amenable groups, and led Genevois and Tessera to introduce the scaling group Sc(X) of an amenable bounded degree graph X. The main result of our article is a rigidity property for quasi-isometries between lamplighters with lamps of polynomial growth. Under assumptions on G and H, any such quasi-isometry N\wr G\longrightarrow M\wr H must be measure-scaling for some scaling factor depending on the growth degrees of N and M. In particular, the scaling group of such wreath products is reduced to \lbrace 1\rbrace. As applications, we obtain additional examples of pairs of quasi-isometric groups that are not biLipschitz equivalent. We also give applications to the quasi-isometric classification of some iterated wreath products, and we exhibit the first example of an amenable finitely generated group H which is lamplighter-rigid, in the sense that ℤ/nℤ\wr H and ℤ/mℤ\wr H are quasi-isometric if and only if n=m.