Simplicial volume, barycenter method, and bounded cohomology

Abstract

We prove that the locally finite simplicial volume of most \({\mathbb {Q}}\)-rank 1 locally symmetric spaces is positive, which has been open for many years. As a corollary, we improve the degree theorem of Connell and Farb for \({\mathbb {Q}}\)-rank 1 locally symmetric spaces. During the course of a proof, we show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most higher rank symmetric spaces. We also address the issue of surjectivity of the comparison map for semisimple Lie groups of rank 2.