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

TL;DR

This paper explores the homotopy equivalence between unstable regions of Bruhat-Tits buildings and spherical Tits buildings for arithmetic groups over function fields, extending previous results to higher ranks and principal congruence subgroups.

Contribution

It generalizes known homotopy equivalence results to the non-semistable parts of Bruhat-Tits buildings for higher rank groups and principal congruence subgroups over function fields.

Findings

Homotopy equivalence established for non-semistable parts of Bruhat-Tits buildings.

Extension of results to principal congruence subgroups in $GL_r(K)$.

Generalization from $GL_2$ to higher rank groups.

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.