The Center Conjecture for spherical buildings of types F4 and E6

TL;DR

This paper proves a key property of convex subcomplexes in spherical buildings of types F4 and E6, showing they are either subbuildings or have automorphisms fixing a point, using differential geometry and metric space theory.

Contribution

It extends the center conjecture to F4 and E6 types, providing new proofs with differential geometric methods and unifying classical and exceptional types.

Findings

Convex subcomplexes are either subbuildings or have fixed points under automorphisms.

The approach applies differential geometry and metric space theory to spherical buildings.

Provides alternative proofs for classical types using the same techniques.

Abstract

We prove that a convex subcomplex of a spherical building of type F4 or E6 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Our approach is differential-geometric and based on the theory of metric spaces with curvature bounded above. We use these techniques also to give another proof of the same result for the spherical buildings of classical type.