A property of the lamplighter group (Q2309695)

A subgroup \(H\) of a group \(G\) is said to be inert (commensurated in another terminology) if \(H\) and \(H^g=g^{-1}Hg\) are commensurate for all \(g\in G\), meaning that \(H\cap H^g\) always has a finite index in both \(H\) and \(H^g\). It is shown that the inert subgroups of the lamplighter group \(\mathbb{Z}_pwr\mathbb{Z}, p \) is a prime, fall into exactly five commensurability classes. An application of this result to the theory of totally disconnected locally compact groups is given. A dynamical aspect of the property investigated in the paper related to the concept of algebraic entropy is also presented.