Coarse classification of abelian groups and amenable shift-homogeneous   metric spaces
  In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces. 
Taras Banakh, Matija Cencelj, Du\v{s}an Repov\v{s}, Ihor Zarichnyi
math.MG math.GR