Embedding products of trees into higher rank
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916234553982976 |
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| author | Bensaid, Oussama Nguyen, Thang |
| author_facet | Bensaid, Oussama Nguyen, Thang |
| contents | We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of rank $n$ with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Embedding products of trees into higher rank Bensaid, Oussama Nguyen, Thang Group Theory Metric Geometry 20F65, 20F69, 30L05, 53C35 We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of rank $n$ with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings. |
| title | Embedding products of trees into higher rank |
| topic | Group Theory Metric Geometry 20F65, 20F69, 30L05, 53C35 |
| url | https://arxiv.org/abs/2405.02226 |