McMahon, Elise2024-04-052024-04-052023-08McMahon_cornellgrad_0058F_13743http://dissertations.umi.com/cornellgrad:13743https://hdl.handle.net/1813/11470679 pagesThis thesis constructs a spectral sequence that converges to the homology of the discrete group $O(p,q)$ with twisted $\mathbb{Z}[\tfrac{1}{2}]$-coefficients, and uses the spectral sequence to compute the group homology $H_{j} (O(p,q), \mathbb{Z} [ \tfrac{1}{2} ]^{\sigma} ) =0$ for $j < \lceil \tfrac{p+q}{2} \rceil$, along with some other low-degree computations. In particular, for $p+q =4$, this gives rise to a four term exact sequence with first term $H_3 \left(O(p,q), \mathbb{Z}[\tfrac{1}{2}]^{\sigma} \right)$ and final term $H_2 \left(O(p,q), \mathbb{Z}[\tfrac{1}{2}]^{\sigma} \right)$. The spectral sequence arises from the spectral sequence for the total homotopy cofiber of a cube. The particular cube was first constructed by Goncherov, where the vertices are scissors congruence groups and the edges are Dehn invariants. Campbell and Zakharevich constructed a cube of simplicial sets such that after applying homology, it yields the classical scissors congruence groups and Dehn invariants in the case of spherical and hyperbolic geometry.By simplicial set magic, the homotopy cofiber of this cube can be simplified to the group homology of the isometry group, $O(n)$ and $O^+(1,n-1).$ Although the geometry in the case of $O(p,q)$ is pseudo-Riemannian, and so scissors congruence is not even defined, we generalize their methods to construct a similar spectral sequence in the case of $O(p,q).$engroup homologyindefinite orthogonal grouporthogonal groupsimplicial setsspectral sequencedissertation or thesishttps://doi.org/10.7298/595t-qm61