Abstract
Thurston showed that the fundamental group of a closed atoroidal 3-manifold admitting a co-oriented taut foliation acts faithfully on the circle by orientation-preserving homeomorphisms. This action on the circle is called a universal circle action, due to the rich information it carries. In this chapter, we first review Thurston’s theory of universal circles and follow-up work of other authors. We note that the universal circle action of a 3-manifold group always admits an invariant lamination. A group acting on the circle with an invariant lamination is called a laminar group. In the second half of the chapter, we discuss the theory of laminar groups and prove some interesting properties of laminar groups under various conditions.
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Acknowledgements
We thank Michele Triestino, Steven Boyer, Sanghyun Kim, Thierry Barbot and Michel Boileau for fruitful conversations. We would like to give special thanks to Hongtaek Jung for a careful reading of an earlier draft and giving helpful comments regarding Sect. 10.9. Finally we greatly appreciate Athanase Papadopoulos and the anonymous referee for their valuable comments which improved the exposition of this chapter. The first half of the chapter was written based on the mini-course the first author gave in the workshop “Low dimensional actions of 3-manifold groups” at Université de Bourgogne in November, 2019. The first author was partially supported by Samsung Science & Technology Foundation grant No. SSTF-BA1702-01, and the second author was partially supported by the Mid-Career Researcher Program (2018R1A2B6004003) through the National Research Foundation funded by the government of Korea.
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Baik, H., Kim, K. (2020). Laminar Groups and 3-Manifolds. In: Ohshika, K., Papadopoulos, A. (eds) In the Tradition of Thurston. Springer, Cham. https://doi.org/10.1007/978-3-030-55928-1_10
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