Classify finitely generated groups quasi-isometric to the lamplighter Z/2Z wr Z^2

Determine all finitely generated groups that are quasi-isometric to the lamplighter group Z/2Z wr Z^2. The goal is to give a complete characterization of the quasi-isometry class of Z/2Z wr Z^2 within the class of finitely generated groups.

Background

The paper develops a general framework for the large-scale geometry of "halo" products and obtains strong embedding and rigidity results. Nevertheless, a complete quasi-isometric classification of finitely generated groups that are quasi-isometric to halo products is stated to be currently out of reach. As a concrete instance of this broader gap, the authors single out the lamplighter over Z^2 as a key example whose quasi-isometric class remains unknown.

This problem sits at the intersection of geometric group theory and coarse geometry, where identifying all groups quasi-isometric to a given model group can reveal deep structural constraints and potential invariants distinguishing quasi-isometry classes.