Asymptotic cones of snowflake groups and the strong shortcut property
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866918160321478656 |
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| author | Cashen, Christopher H. Hoda, Nima Woodhouse, Daniel J. |
| author_facet | Cashen, Christopher H. Hoda, Nima Woodhouse, Daniel J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_11626 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic cones of snowflake groups and the strong shortcut property Cashen, Christopher H. Hoda, Nima Woodhouse, Daniel J. Group Theory Metric Geometry 20F65, 20F69, 51F30 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. |
| title | Asymptotic cones of snowflake groups and the strong shortcut property |
| topic | Group Theory Metric Geometry 20F65, 20F69, 51F30 |
| url | https://arxiv.org/abs/2202.11626 |