Some noncoherent, nonpositively curved K\"ahler groups

Abstract

If $\Gamma$ is any nonuniform lattice in the group ${\rm PU}(2,1)$, let $\overline{\Gamma}$ be the quotient of $\Gamma$ obtained by filling the cusps of $\Gamma$ (i.e. killing the center of parabolic subgroups). Assuming that such a lattice $\Gamma$ has positive first Betti number, we prove that for any sufficiently deep subgroup of finite index $\Gamma_{1} < \Gamma$, the group $\overline{\Gamma_{1}}$ is noncoherent. The proof relies on previous work of M. Kapovich as well as of C. Hummel and V. Schroeder.