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 \geq 2 (excluding SL(3, \mathbb R )/SO(3) and SL(4, \mathbb R )/SO(4)), the p -Jacobian has uniformly bounded norm, provided p \geq n–r+2 . As a consequence, for the corresponding non-compact, connected, semisimple real Lie group G , in degrees p \geq 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.

Locations

• arXiv (Cornell University) - View - PDF
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