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

Abstract

Let $\Gamma$ be an arithmetic subgroup of $SU(d,1)$ with cusps, and let $X_\Gamma$ be the associated locally symmetric space. 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 $SU(d,1)$ is residually finite. We also give an example of a such a group $\Gamma$ for which $H^1_!(X_\Gamma,\mathbb{C})$ is non-zero.