Bounded characteristic classes and flat bundles

TL;DR

This paper characterizes when certain cohomology classes of flat bundles are bounded, linking this property to the simple connectivity of the radical's derived group and providing equivalent conditions involving stable commutator length and distortion.

Contribution

It establishes a precise criterion for boundedness of characteristic classes in terms of the radical's derived group and related algebraic conditions.

Findings

Bounded classes correspond to simply connected derived radical.

Equivalent conditions involve stable commutator length.

Provides a complete characterization of bounded characteristic classes.

Abstract

Let G be a connected Lie group, G^d the underlying discrete group, and BG, BG^d their classifying spaces. Let R denote the radical of G. We show that all classes in the image of the canonical map in cohomology H^*(BG,R)->H^*(BG^d,R) are bounded if and only if the derived group [R,R] is simply connected. We also give equivalent conditions in terms of stable commutator length and distortion.