Laminations and groups of homeomorphisms of the circle

By Danny Calegari, Nathan M. Dunfield. 2002

174 citations recorded by the source corpus.

Abstract.If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π1(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 π1(M) is isomorphic to a subgroup of Homeo(S1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S1. 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/204f777cb28527099ea4c8eea67458e69f154af5

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Citations in the corpus

Random Conformal Dynamical Systems (2005, methodology): “In the case of a foliation by hyperbolic surfaces of a 3-manifold, Thurston has constructed the circle at infinity (see [ CaD ], [F]).”
On the dynamics of (left) orderable groups (2007, background): “) In the recent years, the possibility of ordering many interesting groups (Thompson’s group F [50], braid groups [18], mapping class groups of punctured surfaces with boundary [58], fundamental groups of some hyperbolic 3-dimensional manifolds [4, 9, 15, 57], etc.”
Foliations, orders, representations, L-spaces and graph manifolds (2014, background): “p of an irreducible rational homology 3-sphere graph manifold which admits a cooriented taut foliation acts by orientation-preserving homeomorphisms on S1 via Thurston’s universal circle construction [CD], and hence is circularly-orderable. One can promote the circular-ordering to a left-ordering wh”
Infinitely many hyperbolic 3-manifolds which contain no Reebless foliation (2003, background): “We note that recently Calegari and Dunﬁeld [CD03] announced that they too can generate examples of closed hyperbolic 3-manifolds containing no taut foliation.”
Monopoles and lens space surgeries (2003, methodology): “) The first examples of hyperbolic three-manifolds which admit no taut foliation were constructed in [37], see also [6] In the interest of concreteness, we include the following:”
Ordered Groups and Topology (2015, methodology): “In [14] this algorithm was used to discover Example 5.”
Foliations and the Geometry of 3-Manifolds (2007, background): “Some (but by no means all) of the content of this theory can be found already in the literature, especially [236], [239], [82], [95], [173], [73], [75], [72], [31], [33], [35], [40] and [37], although I hope my presentation and perspective offers something new, even to the experts.”
On L-spaces and left-orderable fundamental groups (2011, background): “As H1(S 3 1/q(K)) = 0, the foliation is co-oriented and the representation π1(S 3 1/q(K)) → Homeo+(S 1) provided by Thurston’s universal circle construction [5] lifts to H̃omeo+(S 1).”
Orderable 3-manifold groups (2002, background): “Proof Calegari and Dunfield prove [CD] that under the conditions of the corollary, the fundamental group of M admits a faithful representation to Homeo+(S (1)) arising from Thurston’s universal circle construction.”