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.
We discuss properties of the complete Euler characteristic of a group G of
type FP over the complex numbers and we relate it to the L2-Euler
characteristic of the centralizers of the elements of G.
We define a transfer map in the setting of bounded cohomology with certain
metric G-module coefficients. As an application, we extend a theorem of
Chatterji, Mislin, Pittet and Saloff-Coste on the comparison map from
Borel-bounded to Borel cohomology, to cover the case of Lie groups with
finitely many connected components.