Speaker
Description
The talk is based on the author’s paper [1].In the paper [2], Vaserstein showed that for any coefficient ring $C$ of finite Krull dimension and any $r \ge \max(3, \dim C +2)$, an arbitrary matrix from the special linear group over a polynomial ring $g \in SL_r(C[x_1, . . . , x_n])$ can be reduced to the diagonal shape $diag(g’,1)$ where $g’ \in SL_{r−1}(C[x_1, . . . , x_n])$, by a bounded number (namely $n(21n − 79)/2 + 33nr + 4r − 4)$ of elementary operations. He also deduced from this the similar result for the symplectic group.
In the talk we state the similar result recently obtained by the author for the split orthogonal group. That is the last remaining case among the split classical groups. This result can be viewed as the effective version of the early surjective $K_1$-functor stability, proved by Suslin and Kopeiko in [3].
We also discuss the connection of such theorems with the proof of the Kazhdan property (T) for split groups over finitely generated rings.
