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Laminations and groups of homeomorphisms of the circle. (English) Zbl 1025.57018

“If \(M\) is an atoroidal 3-manifold with a taut foliation, Thurston showed that \(\pi_1(M)\) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so \(\pi_1(M)\) is isomorphic to a subgroup of Homeo(\(S^1\)). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on \(S^1\). As a corollary, the Weeks manifold does not admit a tight essential lamination with solid torus guts, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston’s universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem”.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
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