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We study the isoperimetric and spectral profiles of certain families offinitely generated groups defined via actions on labelled Schreier graphs andsimple {\em gluing} of such. In one of our simplest constructions---the {\empocket-extension} of a group $G$---this leads to the study of certain finitelygenerated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$.Some sharp estimates are obtained while many challenging questions remain.

Isoperimetric profiles and random walks on some groups defined by piecewise actions