## Institute for Mathematical Sciences

## Preprint ims06-02

** A. Epstein, V. Markovic, D. Saric**
*Extremal maps of the universal hyperbolic solenoid*

Abstract: 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.

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