the question whether the induced maps on homology groups become isomorphisms for large $k$. This happens for many series of groups, for instance for symmetric groups, for mapping class groups and for a large class of linear groups. Homological stability results for general linear groups have been proved by various authors. The direct limit of the above sequence is called the stable general linear group $\operatorname{GL}(R)$, whose homology is isomorphic to almost all of the homology groups of the unstable groups in the above sequence by the aforementioned homological stability results.

Homological stability is useful since stable homology groups are usually easier to compute. For general linear groups, stable group homology is interesting because of its connection to algebraic $K$-Theory via the Hurewicz map: $$K_i(R) \rightarrow H_i(\operatorname{GL}(R)).$$ A related theory called hermitian $K$-theory is related to the homology of stable unitary groups.

Homological stability results for classical groups have been proved by different authors over the years, usually over rings with additional conditions such as finite stable rank, see [@Knu:HLG:01 Chapter 2] for an overview. In this thesis, we restrict ourselves to classical groups over division rings. These groups have associated spherical buildings whose structure allows us to prove stronger 