On the fundamental groups of the complements of Hurwitz curves

Abstract

It is proved that the commutator subgroup of the fundamental group of the complement of any plane affine irreducible Hurwitz curve (respectively, any plane affine irreducible pseudoholomorphic curve) is finitely presented. It is shown that there exists a pseudo-holomorphic curve (a Hurwitz curve) in $\mathbb C\mathbb P^2$ whose fundamental group of the complement is not Hopfian and, respectively, this group is not residually finite. In addition, it is proved that there exist an irreducible nonsingular algebraic curve $C\subset \mathbb C^2$ and a bi-disk $D\subset \mathbb C^2$ such that the fundamental group $\pi_1(D\setminus C)$ is not Hopfian.