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 $Γ\subset GL_2(K)$, 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.