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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?

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    $\begingroup$ @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. $\endgroup$
    – LSpice
    Commented Apr 28, 2022 at 1:55
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    $\begingroup$ I guess you ask that $\Gamma$ stabilizes $S$ ? $\endgroup$
    – Paul Broussous
    Commented May 6, 2022 at 17:08
  • $\begingroup$ @PaulBroussous, thanks! You are right, and I have edited accordingly. $\endgroup$
    – LSpice
    Commented May 6, 2022 at 17:09

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Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$

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  • $\begingroup$ 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. $\endgroup$
    – LSpice
    Commented Apr 30, 2022 at 10:21
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    $\begingroup$ @LSpice a reference-book for CAT⁡(κ)-spaces: arxiv.org/abs/1903.08539 (sorry for the self-advertisement). $\endgroup$
    – Anton Petrunin
    Commented May 1, 2022 at 21:41
  • $\begingroup$ Re, it's hardly an ad--I asked. Thank you! $\endgroup$
    – LSpice
    Commented May 5, 2022 at 15:17
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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.

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