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

TL;DR

This paper introduces the horospherical limit set for modules over groups acting on CAT(0) spaces, linking geometric group theory, tropical geometry, and invariants like the Bieri-Neumann-Strebel invariant.

Contribution

It defines the horospherical limit set in CAT(0) spaces and explores its connections to various geometric and algebraic invariants, bridging multiple mathematical areas.

Findings

$ ext{Sigma}(M;A)$ can be the set of all conical limit points.

$ ext{Sigma}(M;A)$ relates to the complement of a spherical building.

$ ext{Sigma}(M;A)$ connects to the Thurston norm via Bieri-Neumann-Strebel invariant.

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 Poincar\'e's limit set $\Lambda (\Gamma)$ of a discrete group $\Gamma $ of M\"obius 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.