Some noncoherent, nonpositively curved Kähler groups

Abstract

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