In Vogtmann's paper "Spherical posets and homological stability for $O_{n,n}$" it is shown that for all fields different than the field $F_2$ with two elements the homology groups of the orthogonal group $O_{n,n}$ stabilize. Is there a stability result over the field $F_2$ and also for other orthogonal groups?
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1$\begingroup$ Google "Homology stability for unitary groups" ... results are often proved in this generality. In the split case (i.e. for $\mathrm{O}_{n,n}$ in the orthogonal case), there is a wide class of rings for which stability is known (Mirzaii - Van der Kallen). In the general case (e.g. for $\mathrm{O}_{n,k}$ with $k$ fixed and $n\to \infty$), much less is known (but it holds e.g. for number fields, their completions, and some S-arithmetic rings). You can also read Djament's Sur l'homologie des groupes unitaires à coefficients polynomiaux and see references therein. $\endgroup$– few_repsCommented Nov 24, 2015 at 10:38
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4$\begingroup$ (I don't see why some users felt this question had to be put on hold. It is a natural question in this very degree of broadness.) $\endgroup$– few_repsCommented Nov 24, 2015 at 10:45
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Yes, this follows from Section 5.4 of Randal-Williams--Wahl. See also Remark 1.3 of Sprehn--Wahl.
