Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs

TL;DR

This paper demonstrates that certain homogeneous graphs and spaces are not quasi-isometric to Cayley graphs of finitely generated groups, advancing the understanding of quasi-isometric rigidity and classification of specific geometric structures.

Contribution

It introduces the method of coarse differentiation to prove non-quasi-isometry of specific spaces to Cayley graphs, addressing open questions and conjectures in geometric group theory.

Findings

Certain homogeneous graphs are not quasi-isometric to Cayley graphs.

The method of coarse differentiation is effective in proving non-quasi-isometry.

Results support quasi-isometric rigidity of lattices in Sol and lamplighter groups.

Abstract

In this paper, we prove that certain spaces are not quasi-isometric to Cayley graphs of finitely generated groups. In particular, we answer a question of Woess and prove a conjecture of Diestel and Leader by showing that certain homogeneous graphs are not quasi-isometric to a Cayley graph of a finitely generated group. This paper is the first in a sequence of papers proving results announced in [EFW0]. In particular, this paper contains many steps in the proofs of quasi-isometric rigidity of lattices in Sol and of the quasi-isometry classification of lamplighter groups. The proofs of those results are completed in [EFW1]. The method used here is based on the idea of "coarse differentiation" introduced in [EFW0].