Earthquakes and Thurston’s boundary for the Teichmüller space of the universal hyperbolic solenoid

Dragomir Šarić

Vol. 233 (2007), No. 1, 205–228

DOI: 10.2140/pjm.2007.233.205

Abstract

A measured lamination on the universal hyperbolic solenoid 𝒮 is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on the universal hyperbolic solenoid 𝒮 is uniquely determined by a measured lamination on 𝒮; it is a leafwise earthquake with the leafwise earthquake measure equal to the leafwise measured lamination. Leafwise earthquakes fit together to produce a new hyperbolic metric on 𝒮 which is transversely continuous, and we show that any two hyperbolic metrics on 𝒮 are connected by an earthquake. We also establish the space of projective measured lamination PML(𝒮) as a natural Thurston-type boundary to the Teichmüller space T(𝒮) of the universal hyperbolic solenoid 𝒮. The baseleaf-preserving mapping class group MCG_BLP(𝒮) acts continuously on the closure T(𝒮) ∪ PML(𝒮) of T(𝒮). Moreover, the set of transversely locally constant measured laminations on 𝒮 is dense in ML(𝒮).

Keywords

earthquakes, solenoid, laminations, Thurston’s boundary

Mathematical Subject Classification 2000

Primary: 30F60, 30F45, 32H02, 32G05

Secondary: 30C62

Milestones

Received: 17 October 2006

Accepted: 6 November 2006

Published: 1 November 2007