On some pro-p groups from infinite-dimensional Lie theory

TL;DR

This paper explores pro-p groups derived from infinite-dimensional Lie theory, focusing on their topological properties, simplicity, and linearity questions, by examining completions of subgroups in Kac-Moody groups over finite fields.

Contribution

It introduces a new class of pro-p groups from infinite-dimensional Lie theory and proves their topological finite generation and simplicity, advancing understanding of their algebraic structure.

Findings

Pro-p Sylow subgroups are topologically finitely generated.

Certain Kac-Moody groups are shown to be abstractly simple.

Discussion on the potential non-linearity of these pro-p groups.

Abstract

We initiate the study of some pro-p-groups arising from infinite-dimensional Lie theory. These groups are completions of some subgroups of incomplete Kac-Moody groups over finite fields, with respect to various completions of algebraic or geometric origin. We show topological finite generation for the pro-p Sylow subgroups in many complete Kac-Moody groups. This implies abstract simplicity of the latter groups. We also discuss with the question of (non-)linearity of these pro-p groups.