We prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii-van der Kallen respectively. Combining these results with the machinery introduced by Galatius-Randal-Williams to prove homological stability for moduli spaces of simply-connected manifolds of dimension , we get an extension of their result to the case of virtually polycyclic fundamental groups.
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Nina Friedrich, Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds. Doc. Math. 22 (2017), pp. 1729–1774DOI 10.4171/DM/609