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

摘要

The observation that the zero-dimensional Geometric Invariant Sigma(0) (G; A) of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincare's limit set Lambda(Gamma) of a discrete group Gamma of Mobius transformations (which contains the horospherical limit set of Gamma) to the roots of tropical geometry (closely related to Sigma(0) (G; A) when G is abelian). We explore this trail by introducing the horospherical limit set, Sigma(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 partial derivative M. On the way, we meet instances where Sigma(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).