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

## TL;DR

This paper uses Bruhat-Tits theory to explicitly construct minimal generating sets for maximal pro-p subgroups of semisimple groups over local fields, revealing a linear relation with the root system's rank.

## Contribution

It provides an explicit method to compute minimal generators of maximal pro-p subgroups in quasi-split simply-connected semisimple groups using Bruhat-Tits theory.

## Key findings

- Minimal number of generators is linear in the rank of the root system.
- Explicit parametrizations of maximal tori and root groups enable generator computation.
- The approach applies to groups over local fields with residual characteristic p.

## Abstract

Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to 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 corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of a suitable root system.