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

Abstract

A measured laminations on the universal hyperbolic solenoid $§$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid $§$ is uniquely determined by a measured lamination on $§$; 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 $§$ which is transversely continuous and we show that any two hyperbolic metrics on $§$ are connected by an earthquake. We also establish the space of projective measured lamination $PML(§)$ as a natural Thurston-type boundary to the Teichmüller space $T(§)$ of the universal hyperbolic solenoid $§$. The (baseleaf preserving) mapping class group $MCG_{BLP}(§)$ acts continuously on the closure $T(§)\cup PML(§)$ of $T(§)$. Moreover, the set of transversely locally constant measured laminations on $§$ is dense in $ML(§)$.