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

TL;DR

This paper investigates the residual finiteness of certain extensions of arithmetic subgroups of SU(d,1) with cusps, linking it to the non-vanishing of a specific cohomology group and providing explicit examples.

Contribution

It establishes a connection between non-zero first inner cohomology and residual finiteness of pre-images in connected covers of SU(d,1), and provides explicit examples.

Findings

Residual finiteness holds when $H^1_!$ is non-zero.

Constructs explicit examples of groups with non-zero $H^1_!$.

Links cohomology properties to residual finiteness in arithmetic groups.

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.