a tool at hand. This enables us to construct a spectral sequence which can be used for homological stability results for various series of groups.

More concretely, using buildings, we develop a method to prove homological stability for a large class of groups: those which act strongly transitively on a weak spherical building. Among these groups are general and special linear groups as well as unitary, symplectic and orthogonal groups. We apply our method to concrete series of groups to obtain homological stability results which improve the previously known stability ranges in many cases.

Homological stability proofs are usually based on the following idea: Consider the action of a group on a highly connected simplicial complex, find smaller groups of the same series as stabiliser subgroups and inspect the right spectral sequence.

The starting point for our investigation are the homological stability proofs for linear, symplectic and orthogonal groups by Charney in [@Cha:HSD:80] and [@Cha:gtV:87] and by Vogtmann in [@Vog:HSO:79] and [@Vog:SPH:81]. Our method is based on the observation that the simplicial complexes used there are closely related to the theory of buildings — they are the *opposition complexes* studied by von Heydebreck in [@vH:HPC:03].

The opposition complexes admit Levi 