The focus of the thesis is to develop a notion of translations in the setting where usually translations are not defined and used that to build interesting geometrical constructions, coherent with the geometry underlying Grushin spaces. The aim being to use these constructions for future applications to homogenisation problems in perforated domains in Grushin spaces. The first step towards the study of homogenisation problems will be to prove a Poincar´e inequality for those perforated domains in the Grushin plane