Earthquakes and Thurston's boundary for the Teichm\"uller 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 Teichm\"uller space T() of the universal hyperbolic solenoid . The (baseleaf preserving) mapping class group MCGBLP() acts continuously on the closure T() PML() of T(). Moreover, the set of transversely locally constant measured laminations on is dense in ML().
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