Relative homology of arithmetic subgroups of SU(3)

Abstract

Let 𝑘 be a global field of positive characteristic. Let G = SU ⁢ ( 3 ) \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–Poincaré characteristic, the relative homology groups of certain arithmetic subgroups 𝐺 of G ⁢ ( k ) \mathcal{G}(k) modulo a representative system 𝔘 of the conjugacy classes of their maximal unipotent subgroups. In other words, we measure how far the homology groups of 𝐺 are from being the coproducts of the corresponding homology groups of the subgroups U ∈ U U\in\mathfrak{U} .