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arXiv · 2603.09754

The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields

Gebhard Böckle·Sriram Chinthalagiri Venkata
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Abstract

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 $\Gamma \subset GL_2(K)$, the $\Gamma$-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 $\Gamma$-unstable region, for $\Gamma \subset GL_r(K)$ a principal congruence subgroup.

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Gebhard Böckle, Sriram Chinthalagiri Venkata. 2026-03-10. The unstable complex in Bruhat-Tits buildings for arithmetic groups over function fields. https://arxiv.org/abs/2603.09754

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