Equivariant measurable liftings

TL;DR

This paper explores the existence and explicit construction of equivariant measurable liftings of functions under group actions, with applications to cocycles and characteristic classes in geometric contexts.

Contribution

It establishes existence results for equivariant liftings when groups act amenably and provides explicit constructions for semi-simple Lie groups using Fatou convergence.

Findings

Existence of equivariant liftings for amenable group actions.

Explicit liftings constructed for semi-simple Lie groups.

Applications to cocycles and characteristic classes.

Abstract

We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Moebius group of the projective line. Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semi-simple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to cocycles for characteristic classes.