A geometric criterion for the boundedness of characteristic classes

We show that for a connected Lie group G, the linearity of its radical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sqrt G}$$\end{document} (that is of its biggest connected normal solvable subgroup), is a necessary and sufficient condition for the boundedness of all Borel cohomology classes of G with integer coefficients, and that the linearity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sqrt G}$$\end{document} is also equivalent to a large-scale geometric property of G (involving distortion).