  We show that the set of points in the Teichmuller space of the universal
hyperbolic solenoid which do not have a Teichmuller extremal representative is
generic (that is, its complement is the set of the first kind in the sense of
Baire). This is in sharp contrast with the Teichmuller space of a Riemann
surface where at least an open, dense subset has Teichmuller extremal
representatives. In addition, we provide a sufficient criteria for the
existence of Teichmuller extremal representatives in the given homotopy class.
These results indicate that there is an interesting theory of extremal (and
uniquely extremal) quasiconformal mappings on hyperbolic solenoids.
