An extension criterion for lattice actions on the circle

TL;DR

This paper provides a precise criterion based on bounded cohomology for when lattice actions on the circle can be extended to the entire group, unifying several rigidity results.

Contribution

It introduces a necessary and sufficient condition using the real bounded Euler class for extending lattice actions on the circle to the ambient group.

Findings

Characterization of extendability via the bounded Euler class

Unification of existing rigidity theorems

Application of bounded cohomology vanishing theorems

Abstract

We establish a necessary and sufficient condition for an action of a lattice by homeomorphisms of the circle to extend continuously to the ambient locally compact group. This condition is expressed in terms of the real bounded Euler class of this action. Combined with classical vanishing theorems in bounded cohomology, one recovers rigidity results of Ghys, Witte--Zimmer, Navas and Bader--Furman--Shaker in a unified manner.