Barycentric straightening and bounded cohomology

In this talk, I will report on joint work with Jean Lafont. We show that, for an $n$-dimensional irreducible symmetric space of rank $r\geq 2$ (excluding $SL(3,\mathbb R)/SO(3)$ and $SL(4, \mathbb R)/SO(4)$), the $p$-Jacobian of barycentrically straightened simplices has uniformly bounded norm, provided $p\geq n-r+2$. As a consequence, for the corresponding non-compact, connected, semisimple real Lie group $G$, 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. I will also discuss some of my recent work on its generalization.