Some noncoherent, nonpositively curved K\"ahler groups

If $\mathrm{\Gamma }$ is any nonuniform lattice in the group $\mathrm{P}\mathrm{U}(2,1)$, let $\bar{\mathrm{\Gamma }}$ be the quotient of $\mathrm{\Gamma }$ obtained by filling the cusps of $\mathrm{\Gamma }$ (i.e. killing the center of parabolic subgroups). Assuming that such a lattice $\mathrm{\Gamma }$ has positive first Betti number, we prove that for any sufficiently deep subgroup of finite index ${\mathrm{\Gamma }}_1<\mathrm{\Gamma }$, the group $\overline{{\mathrm{\Gamma }}_1}$ is noncoherent. The proof relies on previous work of M. Kapovich as well as of C. Hummel and V. Schroeder.