Relative homology of arithmetic subgroups of $\mathrm{SU}(3)$

Abstract

Let $\mathcal{C}$ be a smooth, projective and geometrically integral curve defined over a finite field $\mathbb{F}$. Let $A$ be the ring of function of $\mathcal{C}$ that are regular outside a closed point $P$ and let $k=\mathrm{Quot}(A)$. Let $\mathcal{G}=\mathrm{SU}(3)$ be the non-split group-scheme defined from an (isotropic) hermitian form in three variables. In this work, we describe, in terms of the Euler-Poincar\'e characteristic, the relative homology groups of certain arithmetic subgroups $G$ of $\mathcal{G}(A)$ modulo a representative system $\mathfrak{U}$ of the conjugacy classes of their maximal unipotent subgroups. In other words, we measure how far are the homology groups of $G$ from being the coproducts of the corresponding homology groups of the subgroups $U \in \mathfrak{U}$.