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 Thurstontype 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 $ML(§)$.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 October 2006
 arXiv:
 arXiv:math/0610496
 Bibcode:
 2006math.....10496S
 Keywords:

 Mathematics  Complex Variables;
 Mathematics  Geometric Topology;
 30F60
 EPrint:
 19 pages