Solutions to two open problems in geometric group theory
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909944869027840 |
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| author | Sahattchieve, Jordan A. |
| author_facet | Sahattchieve, Jordan A. |
| contents | We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that, with the usual action of Fm x Zn on the metric product of the Cayley graph of Fm with Rn, every quasiconvex subgroup of Fm x Zn is convex. This answers the question whether a quasiconvex subgroup of a CAT(0) group is a CAT(0) group in the affirmative for the groups F m x Zn. We also prove bounded packing in a special class of polycyclic groups, and we introduce the notion of coset growth and provide a bound for the coset growth of uniform lattices in Sol. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions to two open problems in geometric group theory Sahattchieve, Jordan A. Group Theory 20F65, 20F67, 20F16, 20F19 We introduce a method for analyzing the convex hull of a set in non-positively curved piecewise Euclidean polygonal complexes and we apply this method to prove that, with the usual action of Fm x Zn on the metric product of the Cayley graph of Fm with Rn, every quasiconvex subgroup of Fm x Zn is convex. This answers the question whether a quasiconvex subgroup of a CAT(0) group is a CAT(0) group in the affirmative for the groups F m x Zn. We also prove bounded packing in a special class of polycyclic groups, and we introduce the notion of coset growth and provide a bound for the coset growth of uniform lattices in Sol. |
| title | Solutions to two open problems in geometric group theory |
| topic | Group Theory 20F65, 20F67, 20F16, 20F19 |
| url | https://arxiv.org/abs/2505.07873 |