$R$-equivalence in adjoint classical groups over fields of virtual cohomological dimension $2$

Abstract

Let $F$ be a field of characteristic not $2$ whose virtual cohomological dimension is at most $2$. Let $G$ be a semisimple group of adjoint type defined over $F$. Let $RG(F)$ denote the normal subgroup of $G(F)$ consisting of elements $R$-equivalent to identity. We show that if $G$ is of classical type not containing a factor of type $D_n$, $G(F)/RG(F) = 0$. If $G$ is a simple classical adjoint group of type $D_n$, we show that if $F$ and its multi-quadratic extensions satisfy strong approximation property, then $G(F)/RG(F) = 0$. This leads to a new proof of the $R$-triviality of $F$-rational points of adjoint classical groups defined over number fields.