general linear groups appear in the spectral sequence. This allows us to apply a strong theorem by Sah in [@Sah:HcL:86] on homological stability for general linear groups to prove homological stability for special linear groups.

If $D$ is an infinite field, then $n\geq 2k-1$ implies $$H_k(\operatorname{SL}_{n+1}(D),\operatorname{SL}_{n}(D);{\mathbb Z})=0.$$

For fields of characteristic zero, there is a far better result by Hutchinson and Tao in [@HT:HSS:08] with stability range $n\geq k$. Up to now, the best result known to the author applicable to other infinite fields has been a result by van der Kallen in [@vdK:HSL:80] for rings with stable rank one. It guarantees a stability range of $n\geq 2k$.

Vogtmann originally used a version of this construction to prove homological stability for orthogonal and symplectic groups in [@Vog:HSO:79] and [@Vog:SPH:81]. In this thesis, we investigate the general situation of unitary groups associated to a hermitian form of Witt index $n+1$ on a vector space $V$. This vector space then splits non-canonically as an orthogonal sum of a hyperbolic module ${\mathcal H}_{n+1}$ and an anisotropic complement $E$. We consider the unitary group induced on the subspace ${\mathcal H}_n\perp E$, where ${\mathcal H}_n$ is a hyperbolic module of rank $n$ and ask for 