Transverse surfaces and pseudo-Anosov flows
-
Michael P. Landry
,
Yair N. Minsky
and
Samuel J. Taylor
Abstract
Let 𝜑 be a transitive pseudo-Anosov flow on an oriented, compact 3-manifold 𝑀, possibly with toral boundary. We characterize the surfaces in 𝑀 that are (almost) transverse to 𝜑. When 𝜑 has no perfect fits (e.g. 𝜑 is the suspension flow of a pseudo-Anosov homeomorphism), we prove that any Thurston norm-minimizing surface 𝑆 that pairs nonnegatively with the closed orbits of 𝜑 is almost transverse to 𝜑, up to isotopy. This answers a question of Cooper–Long–Reid. Our main tool is a correspondence between surfaces that are almost transverse to 𝜑 and those that are relatively carried by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher’s Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces.
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2013073
Award Identifier / Grant number: DMS-2005328
Award Identifier / Grant number: DMS-2102018
Funding statement: Michael P. Landry was partially supported by NSF postdoctoral fellowship DMS-2013073. Yair N. Minsky was partially supported by DMS-2005328. Samuel J. Taylor was partially supported by DMS-2102018 and a Sloan Research Fellowship.
Acknowledgements
We thank Chi Cheuk Tsang for his detailed comments on an earlier draft and the anonymous referee for their helpful comments and corrections.
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Articles in the same Issue
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- A new energy method for shortening and straightening complete curves
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- Gradient estimates for the conductivity problem with imperfect bonding interfaces
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- Transverse surfaces and pseudo-Anosov flows
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Articles in the same Issue
- Frontmatter
- A new energy method for shortening and straightening complete curves
- Exceptional Tannaka groups only arise from cubic threefolds
- Gradient estimates for the conductivity problem with imperfect bonding interfaces
- On the structure of singularities of weak mean curvature flows with mean curvature bounds
- Second adjointness and cuspidal supports at the categorical level
- Transverse surfaces and pseudo-Anosov flows
- On the minimal number of closed geodesics on positively curved Finsler spheres
- On the geometry of spaces of filtrations on local rings
