A property of the lamplighter group

TL;DR

This paper classifies inert subgroups of the lamplighter group into five classes and connects these findings with theories of locally compact groups and algebraic entropy.

Contribution

It provides a complete classification of inert subgroups in the lamplighter group and links this classification to broader mathematical theories.

Findings

Inert subgroups fall into exactly five commensurability classes.

The classification connects to totally disconnected locally compact groups.

The results relate to algebraic entropy theory.

Abstract

We show that the inert subgroups of the lamplighter group fall into exactly five commensurability classes. The result is then connected with the theory of totally disconnected locally compact groups and with algebraic entropy.