The quasi-isometry invariance of the Coset Intersection Complex
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917292173950976 |
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| author | Abbott, Carolyn Martínez-Pedroza, Eduardo |
| author_facet | Abbott, Carolyn Martínez-Pedroza, Eduardo |
| contents | For a pair $(G,\mathcal{P})$ consisting of a group and finite collection of subgroups, we introduce a simplicial $G$-complex $\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair $(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as having finite height, having finite width, being almost malnormal, admiting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $\mathcal{P}$ in $G$ are quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Extension graph; in particular, it is quasi-isometric to a tree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_16628 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The quasi-isometry invariance of the Coset Intersection Complex Abbott, Carolyn Martínez-Pedroza, Eduardo Group Theory Geometric Topology For a pair $(G,\mathcal{P})$ consisting of a group and finite collection of subgroups, we introduce a simplicial $G$-complex $\mathcal{K}(G,\mathcal{P})$ called the coset intersection complex. We prove that the quasi-isometry type and the homotopy type of $\mathcal{K}(G,\mathcal{P})$ are quasi-isometric invariants of the group pair $(G,\mathcal{P})$. Classical properties of $\mathcal{P}$ in $G$ correspond to topological or geometric properties of $\mathcal{K}(G,\mathcal{P})$, such as having finite height, having finite width, being almost malnormal, admiting a malnormal core, or having thickness of order one. As applications, we obtain that a number of algebraic properties of $\mathcal{P}$ in $G$ are quasi-isometry invariants of the pair $(G,\mathcal{P})$. For a certain class of right-angled Artin groups and their maximal parabolic subgroups, we show that the complex $\mathcal{K}(G,\mathcal{P})$ is quasi-isometric to the Extension graph; in particular, it is quasi-isometric to a tree. |
| title | The quasi-isometry invariance of the Coset Intersection Complex |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2404.16628 |