Some noncoherent, nonpositively curved Kähler groups

Abstract

If $\Gamma$ is any nonuniform lattice in the group 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.