The geometry of subgroup embeddings and asymptotic cones
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910955892375552 |
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| author | Jarnevic, Andy |
| author_facet | Jarnevic, Andy |
| contents | Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $ω$, we consider a natural subspace, $Cone^ω_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$. Informally, this subspace consists of the points of the asymptotic cone of $G$ represented by elements of the ultrapower $H^ω$. We show that the connectedness and convexity of $Cone^ω_{G}(H)$ detect natural properties of the embedding of $H$ in $G$. We begin by defining a generalization of the distortion function and show that this function determines whether $Cone^ω_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^ω_{G}(H)$ in the asymptotic cone of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_08312 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The geometry of subgroup embeddings and asymptotic cones Jarnevic, Andy Group Theory 20F65 Given a finitely generated subgroup $H$ of a finitely generated group $G$ and a non-principal ultrafilter $ω$, we consider a natural subspace, $Cone^ω_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$. Informally, this subspace consists of the points of the asymptotic cone of $G$ represented by elements of the ultrapower $H^ω$. We show that the connectedness and convexity of $Cone^ω_{G}(H)$ detect natural properties of the embedding of $H$ in $G$. We begin by defining a generalization of the distortion function and show that this function determines whether $Cone^ω_{G}(H)$ is connected. We then show that whether $H$ is strongly quasi-convex in $G$ is detected by a natural convexity property of $Cone^ω_{G}(H)$ in the asymptotic cone of $G$. |
| title | The geometry of subgroup embeddings and asymptotic cones |
| topic | Group Theory 20F65 |
| url | https://arxiv.org/abs/2201.08312 |