of $2$-spherical type.

Let $G$ be a group of Kac-Moody type and assume that its root datum is of $2$-spherical type. Let $G^\dagger$ be the little projective group and let $\hat G$ be the associated Steinberg group in the sense of Tits. Let ${\mathcal W}(G)$ be the associated Wagoner complex.

Then, in almost all cases $$\begin{aligned}
                \pi_0(|{\mathcal W}(G)|) &\cong G/G^\dagger. \\
                \intertext{If $G$ is of non-spherical type or $\operatorname{rank}(G)\geq 3$, we have}
                \pi_1(|{\mathcal W}(G)|) &\cong \ker(\hat G\twoheadrightarrow G^\dagger).
        \end{aligned}$$ By a recent result by Caprace in [@Cap:O2S:07], this means $\pi_1(|{\mathcal W}(G)|)\cong H_2(G^\dagger;{\mathbb Z})$ except over very small fields.

For a precise statement of the above theorem see Theorem \[th:wag\_main\_result\] and Corollary \[cor:wag\_main\_result\].

In addition to this result, we also make the connection to algebraic $K$-theory explicit in Section \[sec:ktheory\]. Finally, we give a definition of affine Wagoner complexes associated to groups admitting root data with valuations in Section \[sec:affine\].

Homological stability {#sec:stability_results}
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There are many homological stability results for various series of classical groups over rings with finite stable rank. Usually, these proofs are tailored to the specific series of groups. In this thesis, we restrict ourselves to division rings in order to have the theory of buildings as 