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Asymptotic cones of
snowflake groups and the strong shortcut property
Christopher H Cashen, Nima Hoda and Daniel J Woodhouse |
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Algebraic & Geometric Topology 25 (2025) 3429–3501
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Abstract |
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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. |
Keywords
snowflake group, asymptotic cone, shortcut group, strongly
shortcut group
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Mathematical Subject Classification
Primary: 20F65, 20F69, 51F30
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References |
Publication
Received: 1 March 2023
Revised: 1 February 2024
Accepted: 24 March 2024
Published: 1 October 2025
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Authors |
| Christopher H Cashen | |
| Faculty of Mathematics University of Vienna Vienna Austria |
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| Nima Hoda | |
| Department of Mathematics Cornell University Ithaca, NY United States |
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| Daniel J Woodhouse | |
| Mathematical Institute University of Oxford Oxford United Kingdom |
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| © 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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