Homological stability for automorphisms of symmetric bilinear forms

We establish homological stability for automorphisms of symmetric bilinear forms over a class of principal ideal domains that includes all fields, the integers, the Gaussian integers, and the Eisenstein integers. In conjunction with Grothendieck-Witt theoretic calculations, this determines a large part of the stable cohomology of the odd orthogonal groups $O_{⟨g,g⟩}(\mathbb{Z})$ in low degrees.