The Teichmüller theory of the solenoid

Abstract

The compact solenoid was introduced by Sullivan in the 1990s as a universal object in the category of pointed unbranched coverings of a closed surface of genus at least two. It can be seen as a “universal closed surface”. Sullivan studied the compact solenoid in relation with his approach to the Ehrenpreis Conjecture, and this object turned out to be equipped with a variety of interseting structures. This chapter is a survey on the theory of the solenoid that includes its Teichmüller space equipped with its Teichmüller metric, its Teichmüller extremal maps, its modular group action, its decorated theory, with generalizations to the case of the non-compact solenoid.