Euclidean earthquakes†

Abstract

William Thurston developed shearing earthquake maps as transformations in the Teichmüller space of hyperbolic Riemann surfaces, and Curtis McMullen expanded these transformations to include grafting earthquake maps on hyperbolic surfaces. These earthquake maps have been extensively studied, but the literature is confined to earthquake transformations on hyperbolic surfaces. In this article, we define similar transformations on the Teichmüller space of compact tori and describe the differences between the hyperbolic and Euclidean earthquakes. †The opinions expressed in this article are solely those of the authors and not the US Air Force or the Department of Defense.