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

Abstract

The observation that the 0-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 Poincaré's limit set $\Lambda (\Gamma )$ of a discrete group $\Gamma$ of Möbius 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 of M. On the way we meet instances where $\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.

This is the final published version

References in corpus (1)

• Expansive subdynamics for algebraic $Z^d$-actions