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 Izv. RAN. Ser. Mat., 1997, Volume 61, Issue 1, Pages 89–112 (Mi izv106)

On the fundamental groups of complements of toral curves

Vik. S. Kulikov

Moscow State University of Railway Communications

Abstract: We show that for almost all curves $D$ in $\mathbb C^2$ given by an equation of the form $g(x,y)^a+h(x,y)^b=0$, where $a>1$ and $b>1$ are coprime integers, the fundamental group of the complement of the curve has presentation $\pi_1(\mathbb C^2 \setminus D) \simeq (x_1,x_2\mid x_1^a=x_2^b)$, that is, it coincides with the group of the torus knot $K_{a,b}$. In the projective case, for almost every curve $\overline D$ in $\mathbb P^2$ which is the projective closure of a curve in $\mathbb C^2$ given by an equation of the form $g(x,y)^a+h(x,y)^b=0$, the fundamental group $\pi_1(\mathbb P^2\setminus\overline D)$ of the complement is a free product with amalgamated subgroup of two cyclic groups of finite order. In particular, for the general curve $\overline D\subset\mathbb P^2$ given by the equation $l_{bc}^a(z_0,z_1,z_2)+l_{ac}^b(z_0,z_1,z_2)=0$, where $l_q$ is a homogenous polynomial of degree $q$, we have $\pi_1(\mathbb P^2\setminus\overline D)\simeq\langle x_1,x_2\mid x_1^a=x_2^b,x_1^{ac}=1\rangle$.

DOI: https://doi.org/10.4213/im106

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English version:
Izvestiya: Mathematics, 1997, 61:1, 89–112

Bibliographic databases:

Document Type: Article
MSC: Primary 14H30; Secondary 14F35, 14H45, 57M05

Citation: Vik. S. Kulikov, “On the fundamental groups of complements of toral curves”, Izv. RAN. Ser. Mat., 61:1 (1997), 89–112; Izv. Math., 61:1 (1997), 89–112

Citation in format AMSBIB
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\by Vik.~S.~Kulikov
\paper On the fundamental groups of complements of toral curves
\jour Izv. RAN. Ser. Mat.
\yr 1997
\vol 61
\issue 1
\pages 89--112
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\crossref{https://doi.org/10.4213/im106}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1440314}
\zmath{https://zbmath.org/?q=an:0907.14013}
\transl
\jour Izv. Math.
\yr 1997
\vol 61
\issue 1
\pages 89--112
\crossref{https://doi.org/10.1070/IM1997v061n01ABEH000106}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33747020665}

• http://mi.mathnet.ru/eng/izv106
• https://doi.org/10.4213/im106
• http://mi.mathnet.ru/eng/izv/v61/i1/p89

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This publication is cited in the following articles:
1. Vik. S. Kulikov, “Alexander modules of irreducible $C$-groups”, Izv. Math., 72:2 (2008), 305–344
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