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The observation that the zero‐dimensional Geometric InvariantΣ   0    (  G  ;  A  )of Bieri–Neumann–Strebel–Renz can be interpreted as a horospherical limit set opens a direct trail from Poincaréˈs limit set    Λ  (  Γ  )    of a discrete group   Γ   of Möbius transformations (which contains the horospherical limit set of   Γ  ) to the roots of tropical geometry (closely related toΣ   0    (  G  ;  A  )when   G   is abelian). We explore this trail by introducing the horospherical limit set,    Σ  (  M  ;  A  )   , of a   G  ‐module   A   when   G   acts by isometries on a proper    CAT  (  0  )    metric space   M  . This is a subset of the boundary at infinity    ∂  M   . On the way, we meet instances where    Σ  (  M  ;  A  )    is the set of all conical limit points (  G   geometrically finite and   M   hyperbolic), the complement of the radial projection of a tropical variety (  G   abelian and   M   Euclidean) or the complement of a spherical building (  G   arithmetic and   M   symmetric).

Limit sets for modules over groups on CAT (0) spaces: from the Euclidean to the hyperbolic