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

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

2006-10-16

arxiv.org/abs/math/0610496v1

Mathematics

Complex Variables

19 pages

Scientific paper

A measured laminations on the universal hyperbolic solenoid $\S$ is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal 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 Teichm\"uller space $T(\S)$ of the universal hyperbolic solenoid $\S$. The (baseleaf preserving) mapping class group $MCG_{BLP}(\S)$ acts continuously on the closure $T(\S)\cup PML(\S)$ of $T(\S)$. Moreover, the set of transversely locally constant measured laminations on $\S$ is dense in $ML(\S)$.

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