We describe a criterion for a real or complex hyperbolic lattice to admit a RFRS tower that consists entirely of congruence subgroups. We use this to show that certain Bianchi groups $\mathrm{ PSL}(\mathcal{ O}_d)$ are virtually fibered on congruence subgroups, and also exhibit the first examples of RFRS K\"ahler groups that are not a subgroup of a product of surface groups and abelian groups.
Categories: math.GT math.GR math.NT