Lampshufflers are semi-direct products having a geometry close to the one of lamplighters. in geometric group theory, they are also a source of examples of groups with unexpected and exotic behaviours.

I will present an ongoing joint work with Corentin Correia, in which we study these groups from an analytical, geometric and measured point of view. In particular, we show a stability property for the existence of orbit equivalence couplings between lampshufflers, and by computing precisely their isoperimetric profiles, extending previous results from Erschler-Zheng and Saloff-Coste-Zheng, we show that our couplings are quantitatively optimal. The computation of the isoperimetric profile also finds applications to the existence problem of quasi-isometries or regular embeddings between lampshufflers, and allows to enrich the classification program initiated recently by Genevois and Tessera.