The asymmetry of Thurston’s earthquake flow

Arana-Herrera, Francisco; Wright, Alex

doi:10.2140/gt.2024.28.2125

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The asymmetry of Thurston's earthquake flow

Francisco Arana-Herrera and Alex Wright

Geometry & Topology 28 (2024) 2125–2144

DOI: 10.2140/gt.2024.28.2125

Abstract

We show that Thurston’s earthquake flow is strongly asymmetric in the sense that its normalizer is as small as possible inside the group of orbifold automorphisms of the bundle of measured geodesic laminations over moduli space. (At the level of Teichmüller space, such automorphisms correspond to homeomorphisms that are equivariant with respect to an automorphism of the mapping class group.) It follows that the earthquake flow does not extend to an $SL (2, ℝ)$ –action of orbifold automorphisms and does not admit continuous renormalization self-symmetries. In particular, it is not conjugate to the Teichmüller horocycle flow via an orbifold map. This contrasts with a number of previous results, most notably Mirzakhani’s theorem that the earthquake and Teichmüller horocycle flows are measurably conjugate.

Keywords

earthquake flow, normalizer, centralizer, dynamics, asymmetry, hyperbolic geometry, Teichmüller theory

Mathematical Subject Classification

Primary: 30F60

Secondary: 32G15

References

Publication

Received: 3 February 2022

Accepted: 25 February 2023

Published: 24 August 2024

Proposed: Anna Wienhard

Seconded: Dmitri Burago, Mladen Bestvina

Authors

Francisco Arana-Herrera
Department of Mathematics University of Maryland College Park, MD United States

Alex Wright
Department of Mathematics University of Michigan Ann Arbor, MI United States

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Volume 28, issue 5 (2024)