On Tits' Centre Conjecture for Fixed Point Subcomplexes

We give a short and uniform proof of a special case of Tits’ Centre Conjecture using a theorem of J-P. Serre and a result from our earlier work. We consider fixed point subcomplexes $X^H$ of the building $X=X(G)$ of a connected reductive algebraic group $G$, where $H$ is a subgroup of $G$.