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

• 成果类型: Article
• 署名作者: Eskin, Alex; Fisher, David; Whyte, Kevin
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2012.176.1.3
• 发表日期: 2012
• 页码: 221-260
• 关键词: large-scale geometry; lipschitz functions; symmetric-spaces; random-walks; RIGIDITY; trees

摘要

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 our 2007 article Quasi-isometries and rigidity of solvable groups. 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 lamp-lighter groups. The proofs of those results are completed in Coarse differentiation of quasi-isometries II; Rigidity for lattices in Sol and Lamplighter groups. The method used here is based on the idea of coarse differentiation introduced in our 2007 article.