Barycentric Straightening and Bounded Cohomology

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 ≥ 2 (excluding S L(3,R)/S O(3) and S L(4,R)/S O(4)), the p-Jacobian has uniformly bounded norm, provided p ≥ n − r + 2. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group G, in degrees p ≥ 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.