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

On the condensation property of the Lamplighter groups and groups of Intermediate growth