Year: 2025
Type: article
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.20F65, 20F69, 51F30 1. Introduction 3429 2. Preliminaries 3433 3. Snowflake groups 3436 4. The snowflake Cayley graph is not strongly shortcut 3448 5.The idea of the proof that asymptotic cones are simply connected 3451 6. Reduction to bi-Lipschitz cycles 3455 7. Bi-Lipschitz cycles in snowflake groups 3457 8. Constructing fillings 3483 9. Proof of the main theorem and closing remarks 3498
Source: Algebraic & Geometric Topology
Institutions: University of Vienna, Cornell University, University of Oxford
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Funder: Austrian Science Fund
OpenAlex ID: W4414962964