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

Let $K$ be a function field in positive characteristic, $\mathrm{\infty }$ be a fixed place of $K$ and $K_{\mathrm{\infty }}$ be the completion of $K$ at $\mathrm{\infty }$. By the work of Serre, it is well known that, for a suitable arithmetic subgroup $\mathrm{\Gamma }\subset G{L}_2(K)$, the $\mathrm{\Gamma }$-unstable region of the Bruhat-Tits tree for $G{L}_2(K_{\mathrm{\infty }})$ 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 $G{L}_r(K_{\mathrm{\infty }})$. Modifying the approach described by Grayson, we are also able show a similar homotopy equivalence for the $\mathrm{\Gamma }$-unstable region, for $\mathrm{\Gamma }\subset G{L}_r(K)$ a principal congruence subgroup.