Abstract
In this article we construct a new motivic measure called the ${\it Jacques}$
${\it Tits}$ ${\it motivic}$ ${\it measure}$. As a first main application of
the Jacques Tits motivic measure, we prove that two Severi-Brauer varieties
(or, more generally, two twisted Grassmannian varieties), associated to
$2$-torsion central simple algebras, have the same class in the Grothendieck
ring of varieties if and only if they are isomorphic. In addition, we prove
that if two Severi-Brauer varieties, associated to central simple algebras of
period $\{3, 4, 5, 6\}$, have the same class in the Grothendieck ring of
varieties, then they are necessarily birational to each other. As a second main
application of the Jacques Tits motivic measure, we prove that two quadric
hypersurfaces (or, more generally, two involution varieties), associated to
quadratic forms of dimension $6$ or to quadratic forms of arbitrary dimension
defined over a base field $k$ with $I^3(k)=0$, have the same class in the
Grothendieck ring of varieties if and only if they are isomorphic. In addition,
we prove that the latter main application also holds for products of quadric
hypersurfaces.
