Quotients of Buildings by Non-uniform Lattices

Abstract

We consider quotients of the Bruhat-Tits building associated to the projective linear groups of dimension $d>2$ over the function field $\mathbb F_q(t)$ by a non-uniform lattice $\Gamma$ which is a congruence subgroup in the non-uniform lattice $ PGL_{d}(R)$, where $R=\mathbb F_q[\frac{1}{t}]$. We determine a fundamental domain and demonstrate that the quotient, while not cofinite, is at least of finite covolume. We do the case $d=3$ in considerable detail.