The Tits Centre Conjecture

Let $\Delta$ be a thick spherical building. A subcomplex $\Sigma$ of $\Delta$ is completely reducible if every simplex of $\Sigma$ has an opposite in $\Sigma$. Let $\Gamma$ be a subgroup of $\operatorname{Aut}(\Delta)$ stabilising $\Sigma$. A simplex fixed by $\Gamma$ is called a simplicial $\Gamma$-centre of $\Sigma$.

Tits Centre Conjecture. If $\Sigma$ is a convex non-completely-reducible subcomplex of $\Delta$, then $\Sigma$ has a simplicial $\Gamma$-centre.

The conjecture is a central result about invariant convex subcomplexes of spherical buildings and was proved in a series of papers.