Some noncoherent, nonpositively curved K\"ahler groups

TL;DR

This paper demonstrates that certain quotients of nonuniform lattices in PU(2,1), obtained by cusp filling, are noncoherent for deep finite index subgroups, revealing new properties of these complex hyperbolic groups.

Contribution

It establishes noncoherence of cusp-filled quotients of nonuniform lattices in PU(2,1) for deep finite index subgroups, extending understanding of their algebraic structure.

Findings

Cusp-filled quotients are noncoherent for deep subgroups.

The result applies to lattices with positive first Betti number.

The proof builds on previous work by Kapovich, Hummel, and Schroeder.

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.