Can an anisotropic reductive group admit a Tits system?

TL;DR

This paper investigates whether anisotropic reductive groups over certain fields can admit split spherical BN-pairs, concluding they are virtually trivial in many cases, thus extending understanding of group structures.

Contribution

It provides the first results showing that anisotropic reductive groups over perfect or local fields cannot admit non-trivial split spherical BN-pairs.

Findings

Anisotropic reductive groups over perfect or local fields have virtually trivial BN-pairs.

Compact groups admit only spherical, virtually trivial BN-pairs if split.

The results extend the understanding of the structure of reductive groups and their BN-pairs.

Abstract

Seeking for a converse to a well-known theorem by Borel-Tits, we address the question whether the group of rational points G(k) of an anisotropic reductive k-group may admit a split spherical BN-pair. We show that if k is a perfect field or a local field, then such a BN-pair must be virtually trivial. We also consider arbitrary compact groups and show that the only abstract BN-pairs they can admit are spherical, and even virtually trivial provided they are split.