Surfaces and Planar Discontinuous Groups

Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter

doi:10.1007/bfb0089692

Cited by 253 publications

(141 citation statements)

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“…Therefore ϕ maps every geometric generator of Γ into a conjugate of a geometric generator of Γ . By Nielsen's theorem (see for instance the first chapter of [13]) we conclude that ϕ is a geometric isomorphism, i.e. it is induced by a homeomorphism which a priori is defined at the singularities.…”

Section: Theorem a Let F And F Be Two Germs Of General Holomorphic Fmentioning

confidence: 92%

Moduli spaces of germs of holomorphic foliations in the plane

Marín

2003

Comment. Math. Helv.

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Abstract. In this paper we study the topological moduli space of some germs of singular holomorphic foliations in (C 2 , 0). We obtain a fully characterization for generic foliations whose vanishing order at the origin is two or three. We give a similar description for a certain subspace in the moduli space of generic germs of homogeneous foliations of any vanishing order and also for generic quasi-homogeneous foliations. In all the cases we identify the fundamental group of these spaces using the Gassner representation of the pure braid group and a suitable holonomy representation of the foliation.

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Section: Theorem a Let F And F Be Two Germs Of General Holomorphic Fmentioning

confidence: 92%

Moduli spaces of germs of holomorphic foliations in the plane

Marín

2003

Comment. Math. Helv.

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“…Now rank(π 1 (F )) = 2g = 4g + 2m − 2 and the rank(π 1 (O)) is at most 2g + 2m − 1 if g > 0 and is 2m − 1 if g = 0 by [12,Theorem 4.16.1]. In any case, the lemma follows.…”

Section: Introduction and Some Examplesmentioning

confidence: 78%

Generalized Hopfian Property, a Minimal Haken Manifold, and Epimorphisms Between 3-Manifold Groups

2002

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We address the question that if π1-surjective maps between closed aspherical 3-manifolds have the same rank on π1 they must be of non-zero degree. The positive answer is proved for Seifert manifolds, which is used in constructing the first known example of minimal Haken manifold. Another motivation is to study epimorphisms of 3-manifold groups via maps of non-zero degree between 3-manifolds. Many examples are given.

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“…By Theorem 4.10.1 of [16], every co-compact planar discontinuous group has a surface group of finite index, where a surface group is simply a fundamental group of some compact orientable surface. Such groups are known to be residually finite (see e.g.…”

Section: Definition 24mentioning

confidence: 99%