Barycentric straightening and bounded cohomology

Abstract

We study the barycentric straightening of simplices in higher rank irreducible symmetric spaces of non-compact type.We show that, for an $n$-dimensional symmetric space of rank $r\ge 2$ (excluding SL(3, $\mathbb{R}$)/SO(3) and SL(4, $\mathbb{R}$)/SO(4)), the $p$-Jacobian has uniformly bounded norm, provided $p\ge n–r+2$. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group $G$, in degrees $p\ge n-r+2$, every degree $p$ cohomology class has a bounded representative. This answers Dupont’s problem in small codimension. We also give examples of symmetric spaces where the barycentrically straightened simplices of dimension $n–r$ have unbounded volume, showing that the range in which we obtain boundedness of the $p$-Jacobian is very close to optimal.