The Center Conjecture for thick spherical buildings

TL;DR

This paper proves Tits' Center Conjecture for thick spherical buildings of types E6, E7, and E8, establishing that convex subcomplexes are either subbuildings or have automorphisms fixing a point.

Contribution

It completes the proof of Tits' Center Conjecture for all thick spherical buildings excluding H4 types, extending previous partial results.

Findings

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

The proof covers types E6, E7, and E8, completing the conjecture for these cases.

The result applies to all thick spherical buildings without H4 factors.

Abstract

In this paper we show that a convex subcomplex of a spherical building of type E6, E7 or E8 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Together with previous results of M\"uhlherr-Tits, and Leeb and the author, this completes the proof of Tits' Center Conjecture for spherical buildings without factors of type H4, in particular, for thick spherical buildings.