Quasi-isometric classification of halo products over Z (two-ended case)

Determine when two halo products over the base group Z are quasi-isometric. Formally, establish necessary and sufficient conditions under which two finitely generated halo groups \mathscr{L} Z and \mathscr{M} Z (associated to halos acting by left-multiplication on Z) are quasi-isometric, thereby addressing the two-ended case where current understanding is incomplete even for lamplighter groups.

Background

The authors’ main results focus on one-ended bases and spaces with the thick bigon property. They note that the case of infinitely-ended groups is not understood even for classical lamplighters, and identify the two-ended case (i.e., over Z) as a natural next step.

Clarifying the quasi-isometry classification over Z would extend the lamplighter/halo paradigm beyond one-ended bases, potentially leveraging techniques from horospherical products and coarse differentiation while adapting the halo framework.