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Finite presentability of the commutator subgroup of the fundamental group of the complement of a plane curve. (English. Russian original) Zbl 0935.14012

Izv. Math. 61, No. 5, 961-967 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 5, 63-70 (1997).
Let \(D\subset \mathbb{C}^{2}\) be an irreducible affine curve whose projective closure \(\overline{D}\subset \mathbb{P}^{2}\) intersects transversally the line at infinity. The author proves that the commutator subgroups: \(N=\left[ G,G\right] \) of the fundamental affine complement group \(G=\pi _{1}\left( \mathbb{C}^{2}\backslash D\right) \), and \(\overline{N}=\left[ \overline{G}, \overline{G}\right] \) of the fundamental projective complement group \( \overline{G}=\pi _{1}\left( \mathbb{P}^{2}\backslash \overline{D}\right) \) are finitely presented. Thus the situation is similar to that of fibred knots. The proof is based on author’s previous work [Vik. S. Kulikov, Russ. Acad. Sci., Izv., Math. 42, No. 1, 67-89 (1994); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 57, No. 1, 76-101 (1993; Zbl 0811.14017)], where he proved that the commutator \(N\) is finitely generated.

MSC:

14F35 Homotopy theory and fundamental groups in algebraic geometry
14H50 Plane and space curves
14H30 Coverings of curves, fundamental group
57M05 Fundamental group, presentations, free differential calculus
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)

Citations:

Zbl 0811.14017
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