Compactifications of Teichmüller spaces

Abstract

We explain three ways of compactifying Teichmüller spaces: the Thurston compactification, the Teichmüller compactification, and the Bers compactification. The mapping class group acts on the Thurston compactification continuously, but neither on the Teichmüller compactification nor on the Bers compactification. We introduce quotient spaces of the Teichmüller boundary and the Bers boundary to which the action of the mapping class group extends continuously, and show that any self-homeomorphism on these quotient spaces is induced from a unique extended mapping class.