2006/08/29 by J. Higes, Higes, J.
Mathematics · #18B30 #20H15 #20M99 #54C55 #54D35 #54D40 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary: 54F45 #Secondary: 54E35
paper · pdf · doi:10.48550/arxiv.math/0608736
openalex publication_date 2006/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the idea of semigroup-controlled asymptotic dimension. This notion generalizes the asymptotic dimension and the asymptotic Assouad-Nagata dimension in the large scale. There are also semigroup controlled dimensions for the small scale and the global scale. Many basic properties of the asymptotic dimension theory are satisfied by a semigroup-controlled asymptotic dimension. We study how these new dimensions could help in the understanding of coarse embeddings and uniform embeddings. In particular we have introduced uncountable many invariants under quasi-isometries and uncountable many bi-Lipschitz invariants. Hurewicz type theorems are generalized and some applications to geometric group theory are shown.