Séance Séminaire

Séminaire Gaston Darboux

A quasi-isometric classification of permutational wreath products

It is in general a hard problem to determine whether two given finitely generated groups are quasi-isometric, even among some "well behaved" classes. One such class, which has been the subject of intensive research in group theory, is the one of wreath products, as they often exhibit unexpected and interesting behaviours. The classification up to quasi-isometry of lamplighters over $\Z$ goes back to 2013, and in a recent work (2021), Genevois and Tessera extended it to all lamplighters over finitely presented one-ended groups. This raises the question of also classifying their permutational variants. In this context, strong rigidity phenomenon as the ones observed for standard wreath products do not hold, whence the need of another approach. After having introduced and discussed several re-inforcements of quasi-isometries, I will try to sketch the main guidelines of the proof of a quasi-isometric classification of some permutational wreath products, that covers a number of classical cases. If time permits, we will also discuss some applications and open problems.