A generalization of a theorem of Vogtmann

Abstract

In 1977, Vogtmann [lo] proved that the groups O,.(k) are homology stable for any field k, k# [F,. That is, the natural homomorphisms H;(O,.(k); Z)-+ H;(O n + I,n+ i(k); Z) are isomorphisms for n sufficiently large. Recently, Betley [4] has extended this theorem to include local rings k. It is generally expected (and occasionally assumed) that the same is true if k=Z and, most likely, for a much larger class of rings as well, but no proof of this exists in the literature. As this fact, for k= Z’, is an essential ingredient in a forthcoming joint paper with R. Lee, I have undertaken to present a proof. The main result contained here is the homology stability of O,.(A) and SpZn(A), with twisted as well as untwisted coefficients, in the case where A is a Dedekind domain. Much of the discussion, however, is carried out in a broader context in the hopes that it can be used for still further generalizations. In particular, it is conjectured that stability holds for any finite algebra A over a commutative ring k with noetherian maximal spectrum, and so as much of the theory as possible is developed in this setting. The bulk of the work, as always in proving homology stability, is the construction of highly connected simplicial complexes on which the groups act with ‘nice’ stabilizer subgroups. In this case, we use the simplicial complex arising from the partially ordered set of ‘hyperbolic unimodular’ sequences (see Section 3) which has the nicest possible stabilizer subgroups, namely lower-dimensional orthogonal or symplectic groups. The proof of the connectedness of this complex occupies Sections l-3. The methods used owe much to the work Maazen [8] and Van der Kallen [7]. The last section, Section 4, contains the stability theorems.