Explicit generators of some pro-p groups via Bruhat-Tits theory

Générateurs explicites de certains sous-groupes pro-p via la théorie de Bruhat-Tits.

Benoit Loisel (1)

1 CMLS - Centre de Mathématiques Laurent Schwartz

Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p subgroups. The maximal pro-p subgroups of quasisplit simply connected semisimple groups can be described in the combinatorial terms of a valued root groups datum, thanks to the Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p subgroups thanks to parametrizations of a suitable maximal torus and of the corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of a suitable root system.

Keywords: profinite groups; buildings; linear algebraic groups; local fields; finite presentations

Domains: Group Theory [math.GR]; Algebraic Geometry [math.AG]

Benoit Loisel. Explicit generators of some pro-p groups via Bruhat-Tits theory. Bulletin de la société mathématique de France, 2021, 149 (2), pp.309-388. ⟨10.24033/bsmf.2831⟩. ⟨hal-01428864v3⟩