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

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 $\Gamma \subset G{L}_2(K)$, the $\Gamma$-unstable region of the Bruhat-Tits tree for $G{L}_2(K_{\infty })$ is naturally homotopy equivalent to the spherical Tits building for $G{L}_2(K)$. Grayson, following Quillen's ideas, generalizes this homotopy equivalence to the non-semistable part of the Bruhat-Tits building for $G{L}_r(K_{\infty })$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $\Gamma$-unstable region, for $\Gamma \subset G{L}_r(K)$ a principal congruence subgroup.

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