3 feb 2016 -- 14:30
Aula di Consiglio, Dip.Matematica, Università "La Sapienza", Roma
Abstract.
A compact Clifford–Klein form of a homogeneous space $G/H$ is a quotient of this homogeneous space by a discrete subgroup $Γ$ of $G$ acting properly discontinuously and cocompactly on $G/H$. When both $G$ and $H$ are semi-simple groups, the action of $G$ on $G/H$ preserves a pseudo-Riemannian metric, and in particular a volume form. I will prove that the volume of the compact Clifford–Klein form $Γ \G/H$ is the integral over a certain “fundamental class” of $Γ$ of some $G$-invariant form $ωH$ on the Riemannian symmetric space $G/K$. As a consequence, one obtains in many cases that this volume is rigid. Moreover, this provides a new obstruction to the existence of compact Clifford–Klein forms of certain homogeneous spaces.