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The geometry of $\mathbb{R}$–covered foliations

Danny Calegari

Geometry & Topology 4 (2000) 457–515

arXiv: math.GT/9903173


We study –covered foliations of 3–manifolds from the point of view of their transverse geometry. For an –covered foliation in an atoroidal 3–manifold M, we show that M̃ can be partially compactified by a canonical cylinder Suniv1 × on which π1(M) acts by elements of Homeo(S1) × Homeo(), where the S1 factor is canonically identified with the circle at infinity of each leaf of ̃. We construct a pair of very full genuine laminations Λ± transverse to each other and to , which bind every leaf of . This pair of laminations can be blown down to give a transverse regulating pseudo-Anosov flow for , analogous to Thurston’s structure theorem for surface bundles over a circle with pseudo-Anosov monodromy.

A corollary of the existence of this structure is that the underlying manifold M is homotopy rigid in the sense that a self-homeomorphism homotopic to the identity is isotopic to the identity. Furthermore, the product structures at infinity are rigid under deformations of the foliation through –covered foliations, in the sense that the representations of π1(M) in Homeo((Suniv1)t) are all conjugate for a family parameterized by t. Another corollary is that the ambient manifold has word-hyperbolic fundamental group.

Finally we speculate on connections between these results and a program to prove the geometrization conjecture for tautly foliated 3–manifolds.

taut foliation, $\mathbb{R}$–covered, genuine lamination, regulating flow, pseudo-Anosov, geometrization
Mathematical Subject Classification 2000
Primary: 57M50, 57R30
Secondary: 53C12
Forward citations
Received: 18 September 1999
Revised: 23 October 2000
Accepted: 14 December 2000
Published: 14 December 2000
Proposed: David Gabai
Seconded: Dieter Kotschick, Walter Neumann
Danny Calegari
Department of Mathematics
Harvard University
Massachusetts 02138