Asymptotic cones of snowflake groups and the strong shortcut property

2022/02/23 by Christopher H. Cashen, Cashen, Christopher H., Nima Hoda +3
Biochemistry, Genetics and Molecular Biology · Mathematics · #20F65 #20F69 #51F30 #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Microtubule and mitosis dynamics

paper · pdf · doi:10.48550/arxiv.2202.11626

openalex publication_date 2022/02/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut. These are the first examples of groups whose asymptotic cones contain `metrically nontrivial' loops but no topologically nontrivial ones.

Related

• Strong shortcuts, generating sets, and isometric circles in asymptotic cones
• Finitely presented groups with infinitely many non-homeomorphic asymptotic cones
• Separable and tree-like asymptotic cones of groups
• A finitely presented group with two non-homeomorphic asymptotic cones
• Groups with non-simply connected asymptotic cones