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Endperiodic automorphisms of surfaces and foliations

Published online by Cambridge University Press:  07 October 2019

JOHN CANTWELL
Affiliation:
Saint Louis University, St. Louis, MO63103, USA email cantwelljc@slu.edu
LAWRENCE CONLON
Affiliation:
Washington University, St. Louis, MO63130, USA email conlonlawrence@icloud.com
SERGIO R. FENLEY
Affiliation:
Florida State University, Tallahassee, FL32306-4510, USA email fenley@math.fsu.edu

Abstract

We extend the unpublished work of Handel and Miller on the classification, up to isotopy, of endperiodic automorphisms of surfaces. We give the Handel–Miller construction of the geodesic laminations, give an axiomatic theory for pseudo-geodesic laminations, show that the geodesic laminations satisfy the axioms, and prove that pseudo-geodesic laminations satisfying our axioms are ambiently isotopic to the geodesic laminations. The axiomatic approach allows us to show that the given endperiodic automorphism is isotopic to a smooth endperiodic automorphism preserving smooth laminations ambiently isotopic to the original ones. Using the axioms, we also prove the ‘transfer theorem’ for foliations of 3-manifolds, namely that, if two depth-one foliations ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime }$ are transverse to a common one-dimensional foliation ${\mathcal{L}}$ whose monodromy on the non-compact leaves of ${\mathcal{F}}$ exhibits the nice dynamics of Handel–Miller theory, then ${\mathcal{L}}$ also induces monodromy on the non-compact leaves of ${\mathcal{F}}^{\prime }$ exhibiting the same nice dynamics. Our theory also applies to surfaces with infinitely many ends.

Type
Original Article
Copyright
© Cambridge University Press, 2019

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