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

摘要

The observation that the 0-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit setopens a direct trail from Poincar\'e's limit set $\Lambda (\Gamma)$ of adiscrete group $\Gamma $ of M\"obius transformations (which contains thehorospherical limit set of $\Gamma $) to the roots of tropical geometry(closely related to $\Sigma ^{0}(G;A)$ when G is abelian). We explore thistrail by introducing the horospherical limit set, $\Sigma (M;A)$, of a G-moduleA when G acts by isometries on a proper CAT(0) metric space M. This is a subsetof the boundary at infinity of M. On the way we meet instances where $\Sigma(M;A)$ is the set of all conical limit points, the complement of a sphericalbuilding, the complement of the radial projection of a tropical variety, or(via the Bieri-Neumann-Strebel invariant) where it is closely related to theThurston norm.