Locally normal subgroups and ends of locally compact Kac-Moody groups

TL;DR

This paper characterizes the large-scale structure of locally normal subgroups in complete Kac-Moody groups over finite fields and shows that one-ended groups are locally indecomposable.

Contribution

It provides a detailed description of locally normal subgroups and proves that one-ended Kac-Moody groups are locally indecomposable under mild conditions.

Findings

Large-scale structure of locally normal subgroups described

One-ended Kac-Moody groups are locally indecomposable

Results depend on properties of the generalised Cartan matrix

Abstract

A locally normal subgroup in a topological group is a subgroup whose normaliser is open. In this paper, we provide a detailed description of the large-scale structure of closed locally normal subgroups of complete Kac-Moody groups over finite fields. Combining that description with the main result from arXiv:2111.07066, we show that under mild assumptions, if the Kac-Moody group is one-ended (a property that is easily determined from the generalised Cartan matrix), then it is locally indecomposable, which means that no open subgroup decomposes as a nontrivial direct product.