Transgression in bounded cohomology and a conjecture of Monod

Abstract

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2, R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles. defined by L κ (α) = κ α.Corollary. The bounded Lefschetz map in (1) is zero in all positive degrees.Returning to Theorem 1, we note that in small degrees n = 3, 4, much stronger vanishing theorems apply: Burger and Monod [17] proved that H 3 cb (G; R) = 0, while Hartnick and the author [28] showed that H 4 cb (G; R) = 0. In large degree, on the other hand, our Theorem 1