subgroups as vertex stabilisers. Using these, we construct a spectral sequence involving relative group homology of Levi subgroups. These Levi subgroups often split as direct of semidirect products of smaller groups which can be used for stability proofs.

We will use this spectral sequence for homological stability proofs for special linear and for unitary groups. For these groups, the Levi subgroups split as direct or semidirect products, both of special linear or unitary groups and, interestingly, of general linear groups. Using strong stability results for general linear groups, we can prove stability results for special linear or unitary groups.

Our method can probably also be used to show low-dimensional homological stability for groups of types $E_6$, $E_7$ and $E_8$. Additionally, one could try to compare group homology of groups of different types using this method. Finally, homological stability results for all reductive algebraic groups should be possible, albeit with a rather weak stability range.

### Homological stability results

The method outlined above has originally been used by Charney in [@Cha:HSD:80] to prove homological stability of general and special linear groups, but yielding a comparatively weak stability range. For special linear groups, however, it is an interesting observation that terms involving 