Totally disconnected locally compact groups with a linear open subgroup

TL;DR

This paper characterizes the structure of totally disconnected locally compact groups with a linear open subgroup and shows that certain simple, locally linear groups are algebraic over local fields.

Contribution

It provides a detailed description of the global structure of these groups and establishes a link to simple algebraic groups over local fields.

Findings

Groups with a linear open subgroup have a specific global structure.

Locally linear, simple, totally disconnected groups are algebraic over local fields.

Abstract

We describe the global structure of totally disconnected locally compact groups having a linear open compact subgroup. Among the applications, we show that if a non-discrete, compactly generated, topologically simple, totally disconnected locally compact group is locally linear, then it is a simple algebraic group over a local field.