Elementary Action of Classical Groups on Unimodular Rows Over Monoid Rings

The elementary action of symplectic and orthogonal groups on unimodular rows of length $2n$ is transitive for $2n\ge \max (4,d+2)$ in the symplectic case, and $2n\ge \max (6,2d+4)$ in the orthogonal case, over monoid rings $R[M]$, where $R$ is a commutative noetherian ring of dimension $d$, and $M$ is commutative cancellative torsion free monoid. As a consequence, one gets the surjective stabilization bound for the $K_1$ for classical groups. This is an extension of J. Gubeladze’s results for linear groups.