The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911505614635008 |
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| author | Böckle, Gebhard Venkata, Sriram Chinthalagiri |
| author_facet | Böckle, Gebhard Venkata, Sriram Chinthalagiri |
| contents | Let $K$ be a function field in positive characteristic, $\infty$ be a fixed place of $K$ and $K_\infty$ be the completion of $K$ at $\infty$. By the work of Serre, it is well known that, for a suitable arithmetic subgroup $Γ\subset GL_2(K)$, the $Γ$-unstable region of the Bruhat-Tits tree for $GL_2(K_\infty)$ is naturally homotopy equivalent to the spherical Tits building for $GL_2(K)$. Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for $GL_r(K_\infty)$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $Γ$-unstable region, for $Γ\subset GL_r(K)$ a principal congruence subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09754 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields Böckle, Gebhard Venkata, Sriram Chinthalagiri Number Theory Let $K$ be a function field in positive characteristic, $\infty$ be a fixed place of $K$ and $K_\infty$ be the completion of $K$ at $\infty$. By the work of Serre, it is well known that, for a suitable arithmetic subgroup $Γ\subset GL_2(K)$, the $Γ$-unstable region of the Bruhat-Tits tree for $GL_2(K_\infty)$ is naturally homotopy equivalent to the spherical Tits building for $GL_2(K)$. Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for $GL_r(K_\infty)$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $Γ$-unstable region, for $Γ\subset GL_r(K)$ a principal congruence subgroup. |
| title | The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.09754 |