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 \(2n\ge \max (4, d+2)\) $2n\ge max(4,d+2)$ in the symplectic case, and \(2n\ge \max (6, 2d+4)\) $2n\ge max(6,2d+4)$ 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 \({\textrm{K}_1}\) ${\text{K}}_1$ for classical groups. This is an extension of J. Gubeladze’s results for linear groups.