On Tits' Centre Conjecture for Fixed Point Subcomplexes

TL;DR

This paper provides a concise, uniform proof of a specific case of Tits' Centre Conjecture for fixed point subcomplexes in buildings associated with reductive algebraic groups, leveraging Serre's theorem and prior results.

Contribution

It offers a new, simplified proof of a special case of Tits' Centre Conjecture using existing theorems, enhancing understanding of fixed point subcomplexes.

Findings

Proved a special case of Tits' Centre Conjecture.

Utilized Serre's theorem and previous work for the proof.

Simplified the proof approach for fixed point subcomplexes.

Abstract

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 $G$.