Quasi-isometric rigidity for lamplighters with lamps of polynomial growth

TL;DR

The paper studies rigidity phenomena for quasi-isometries between lamplighter groups with polynomial-growth lamps, introducing measure-scaling maps and the scaling group to quantify non-bijective large-scale maps. Under amenability and polynomial-growth assumptions on the lamp groups, it proves that any quasi-isometry between N \wr G and M \wr H is measure-scaling with factor $m/n$, where $n$ and $m$ are the growth degrees of $N$ and $M$, respectively, forcing Sc$(N \wr G) = \{1\}$. This yields lamplighter-rigidity results and sharp quasi-isometric classification for iterated wreath products, including new examples of quasi-isometric but non-biLipschitz-equivalent pairs, by exploiting aptolicity of quasi-isometries. The approach relies on an aptolic decomposition of wreath-product quasi-isometries, deriving key constraints on the base-map and lamp-map that translate growth-data into exact scaling factors, thus connecting to and extending prior work GT24b, BGT24, and Whyte's rigidity results for amenable spaces.

Abstract

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 $\mathbb{Z}$ and the work of Dymarz on biLipschitz equivalences of amenable groups, and led Genevois and Tessera to introduce the scaling group $\text{Sc}(X)$ 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 $\mathbb{Z}/n\mathbb{Z}\wr H$ and $\mathbb{Z}/m\mathbb{Z}\wr H$ are quasi-isometric if and only if $n=m$.

Theorems & Definitions (58)

• Definition 1.1
• Theorem 1
• Theorem 2
• Corollary 3
• Theorem 4
• Corollary 5
• Proposition 6
• Proposition 7
• Corollary 8
• Corollary 9