Résume | The classification of certain families of groups up to quasi-isometry has been the object of an intensive study in the last 30 years, culminating with the classification of lattices in Lie groups, and mapping class groups.
The case of amenable, and more specifically solvable groups appears to be much harder, and has known very few developments in comparison. Until recently, only lamplighters over Z, Baumslag-Solitar groups, and very specific polycyclic groups had been treated. With Anthony Genevois, we introduce new tools to study the large scale geometry of certain families of groups obtained as a semi-direct product of a locally finite group with an arbitrary group H: for instance Lamplighter groups (wreath product of H with a finite group), or Lampshuffler groups (semi-direct product of H with permutations finitely supported on H), and other similar constructions.
We prove that if H is one-ended and finitely presented, then any quasi-isometry between any two such groups must preserve the semi-direct product structure in a very strong sense. This yields a complete classification in the case of lamplighters, which partially extend to more general classes. |