# Fixed points on spherical buildings

A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits fixed-point theorem, which, in one of many formulations, states that, if $$\Gamma$$ is a group of isometries of an affine building and $$S$$ is a closed, bounded, convex, $$\Gamma$$-stable subset of the affine building, then $$\Gamma$$ admits a fixed point on $$S$$.

Is there any similar result for spherical buildings (specifically of spherical buildings attached to semisimple groups, in case there are more results for them than for general spherical buildings)? I am particularly interested in results of the form: if $$\Gamma$$ is a group of isometries of a spherical building and $$S$$ is a […] $$\Gamma$$-stable subset of the spherical building, then $$\Gamma$$ admits a fixed point on $$S$$. For example, does this hold if we require $$S$$ to be closed and convex? (Probably not.) Does it hold if we require $$S$$ also to be contractible, or perhaps just never to contain two opposite simplices?

• @AntonPetrunin, thank you! It seems to me that your comment is an answer, so I hope you will make it as such so that I can accept it. Commented Apr 28, 2022 at 1:55
• I guess you ask that $\Gamma$ stabilizes $S$ ? Commented May 6, 2022 at 17:08
• @PaulBroussous, thanks! You are right, and I have edited accordingly. Commented May 6, 2022 at 17:09

Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$
• Thanks! Do you know a good reference for $\operatorname{CAT}(\kappa)$-spaces for someone who is not a geometer? I know a little bit about $\operatorname{CAT}(0)$-spaces because of my work with affine buildings, but I have mostly just encountered them under the general rubric of "spaces of non-positive curvature", not as part of a general family. Commented Apr 30, 2022 at 10:21
The Tits centre conjecture, on which I am hardly an expert and so do not propose to offer a literature survey but which was proven for thick spherical buildings (in particular for spherical buildings of semisimple groups) in Ramos-Cuevas - The center conjecture for thick spherical buildings, says that, if $$S$$ is a convex subcomplex of a spherical building $$B$$, and $$S$$ is not itself a spherical building, then the stabiliser in $$\operatorname{Aut}(B)$$ of $$S$$ admits a fixed point on $$S$$. This seems to answer my question: it holds when $$S$$ is contractible.