Residual finiteness of extensions of arithmetic subgroups of {{,mathrm{SU},}}(d,1) with cusps

Abstract

Let Gamma be an arithmetic subgroup of {{,mathrm{SU},}}(d,1) with cusps, and let X_Gamma be the associated locally symmetric space. In this paper we investigate the pre-image of Gamma in the covering groups of {{,mathrm{SU},}}(d,1). Let H^bullet _!(X_Gamma ,mathbb {C}) be the inner cohomology, i.e. the image in H^bullet (X_Gamma ,mathbb {C}) of the compactly supported cohomology. We prove that if the first inner cohomology group H^1_!(X_Gamma ,mathbb {C}) is non-zero then the pre-image of Gamma in each connected cover of {{,mathrm{SU},}}(d,1) is residually finite. At the end of the paper we give an example of an arithmetic subgroup Gamma satisfying the criterion H^1_!(X_Gamma ,mathbb {C}) ne 0.