Reduced Bers boundaries of Teichm\"uller spaces

TL;DR

This paper introduces the reduced Bers boundary of Teichmüller space, showing its independence from basepoints, the extension of the mapping class group action, and characterizing automorphisms of this boundary.

Contribution

It defines the reduced Bers boundary by collapsing quasi-conformal deformation spaces, proves its invariance under basepoints, and characterizes boundary automorphisms as extended mapping classes.

Findings

Reduced Bers boundary is independent of basepoint.

Mapping class group action extends continuously to the boundary.

Automorphisms of the boundary correspond to extended mapping classes.

Abstract

We consider a quotient space of the Bers boundary of Teichm\"{u}ller space, which we call the reduced Bers boundary, by collapsing each quasi-conformal deformation space into a point. This reduced Bers boundary turns out to be independent of the basepoint, and the action of the mapping class group on the Teichm\"{u}ller space extends continuously to this boundary. We show that every auto-homeomorphism on the reduced Bers boundary comes from an extended mapping class. We also give a way to determine the limit in the reduced Bers boundary up to some ambiguity for parabolic curves for a given sequence in the Teichm\"{u}ller space, by generalising the Thurston compactification.