Surfaces, circles, and solenoids

Previous work of the author has developed coordinates on bundles over the classical Teichmüller spaces of punctured surfaces and on the space of cosets of the Möbius group in the group of orientation-preserving homeomorphisms of the circle, and this work is surveyed here. Joint work with Dragomiř Sari´c is also sketched which extends these results to the setting of the Teichmüller space of the solenoid of a punctured surface, which is defined here in analogy to Dennis Sullivan's original definition for the case of closed surfaces. Because of relations with the classical modular group, the punctured solenoid and its Teichmüller theory have connections with number theory.