Nonabelian K-Theory: The Nilpotent K 1 and General Stability Class of

A functorial filtration GL n = S-1L n 2 S~ - ' ' �9 2 SiLn 2" �9 �9 2 E n of the general linear group GLn, n >/3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S-1L.(A)/S~ is abelian, that S~ _ SILn(A) 2..- is a descending central series, and that SiLn(A) =E#(A) whenever if> the Bass-Serre dimension of A. In particular, the K-functors K1SILn ,= SiL./En are nilpotent for all i ~> 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism SiL~(A)/Si+ILn(A)--*SgL.+I(A)/ S ~+ ~L~ + l(A) is injective whenever n ~> i + 3, so that one has stability results without stability condi- tions, and if A is commutative then S~ agrees with the special linear group SL.(A), so that the functor S~ generalizes the functor SL~ to noncommutative rings. Applying the above to subgroups H of GL~(A), which are normalized by En(A), one obtains that each is contained in a sandwich