The fixed-point conjecture for almost connected locally compact groups on buildings

Let $G$ be an almost connected locally compact group. Recall that a group has property $\operatorname{(FB)}$ when every measurable action of $G$ on a finite-rank building stabilises a spherical residue. Fixed-point conjecture. Every almost connected locally compact group has property $\operatorname{(FB)}$. The theorem proved in the paper establishes this under the additional hypothesis of finite abelian width; the conjecture asserts that this hypothesis is unnecessary in general.