Séminaire Géométrie et groupes discrets

Vanishing simplicial volume for certain affine manifolds

by Prof. Michelle BUCHER (Université de Genève)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Affine manifolds, i.e. manifolds which admit charts given by affine transformations, remain mysterious by the very few explicit examples and their famous open conjectures: the Auslander Conjecture, the Chern Conjecture and the Markus Conjecture. I will discuss an intermediate conjecture, somehow between the Auslander Conjecture and the Chern Conjecture, predicting the vanishing of the simplicial volume of affine manifolds. In a joint work with Chris Connell and Jean-François Lafont, we prove the latter conjecture under some hypothesis, thus providing further evidence for the veracity of the Auslander and Chern Conjectures. To do so, we provide a simple cohomological criterion for aspherical manifolds with normal amenable subgroups in their fundamental group to have vanishing simplicial volume. This answers a special case of a question due to Lück. Joint work with Chris Connell and Jean-François Lafont.