Bounded characteristic classes and flat bundles (Q366945)

Let \(G\) be a connected Lie group and denote by \(BG\) the classifying space of \(G\). If \(BG^\delta\) is the classifying space of the group \(G\) with the discrete topology, a class \(\alpha\) in \(H^n(BG^\delta, A)\) is bounded if it can be represented by a cocycle \(c:G^n\to A\) with bounded image. The authors obtain some conditions for \(G\), under which the classes in the image of \(H^*(BG,\mathbb{R})\to H^*(BG^\delta, \mathbb{R})\) are bounded. Theorem 1.1. Let \(G\) be a connected Lie group and let \(R\) be its radical. The following conditions are equivalent. (1) All elements in the image of \(H^*(BG ,\mathbb{R})\to H^*(BG^\delta, \mathbb{R})\) are bounded. (2) The derived group \([R,R]\) of the radical of \(G\) is simply connected. The authors recall and discuss some background results about Borel cohomology and obtain the Theorem 2.1 where they give some necessary and sufficient conditions in terms of integral Borel cohomology of \(G\) for \(\pi_1(G)\) to be undistorted in \(G\). In Theorem 2.2. some background results related to Theorem 1.1. are given. In section 3, a primary obstruction to the existence of a global section of the universal \(G\)-bundle is obtained. Then the authors prove the existence of a non-zero lower bound on the stable commutator length of all elements in the commutator subgroup of the universal cover of \(G\) that are central and whose stable commutator length is non-zero.