The asymmetry of Thurston's earthquake flow

TL;DR

This paper demonstrates that Thurston's earthquake flow is highly asymmetric, with a minimal normalizer, and cannot be extended or symmetrized via orbifold automorphisms, contrasting with the conjugacy of the Teichmüller horocycle flow.

Contribution

It establishes the strong asymmetry of Thurston's earthquake flow and shows it does not admit continuous symmetries or extensions to SL(2,R)-actions, unlike related flows.

Findings

The normalizer of the earthquake flow is as small as possible.

The earthquake flow cannot extend to an SL(2,R)-action.

It is not conjugate to the Teichmüller horocycle flow via orbifold maps.

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 Teichm\"uller 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 $\mathrm{SL}(2,\mathbf{R})$-action of orbifold automorphisms and does not admit continuous renormalization self-symmetries. In particular, it is not conjugate to the Teichm\"uller horocycle flow via an orbifold map. This contrasts with a number of previous results, most notably Mirzakhani's theorem that the earthquake and Teichm\"uller horocycle flows are measurably conjugate.