  The aim of this short note is to revisit some old results about groups of
intermediate growth and groups of the lamplighter type and to show that the
Lamplighter group $L=\mathbb{Z}_2\wr \mathbb{Z}$ is a condensation group and
has a minimal presentation by generators and relators. The condensation
property is achieved by showing that $L$ belongs to a Cantor subset of the
space $\mathcal{M}_2$ of marked 2-generated groups consisting mostly of groups
of intermediate growth.
