Higher horospherical limit sets for G-modules over CAT(0) spaces

TL;DR

This paper extends the theory of Sigma-invariants for group actions from Euclidean spaces to higher stages in proper CAT(0) spaces, broadening the scope of geometric group theory tools.

Contribution

It introduces a higher-dimensional extension of Sigma-invariants for groups acting on CAT(0) spaces, generalizing previous Euclidean space results.

Findings

Developed higher-stage Sigma-invariants for CAT(0) spaces.

Connected invariants to group actions in more general geometric contexts.

Extended the theoretical framework for analyzing group actions.

Abstract

The Sigma-invariants of Bieri-Neumann-Strebel and Bieri-Renz involve an action of a discrete group G on a geometrically suitable space M. In the early versions, M was always a finite-dimensional Euclidean space on which G acted by translations. A substantial literature exists on this, connecting the invariants to group theory and to tropical geometry (which, actually, Sigma-theory anticipated). More recently, we have generalized these invariants to the case where M is a proper CAT(0) space on which G acts by isometries. The "0th stage" of this was developed in our paper [BG16]. The present paper provides a higher-dimensional extension of the theory to the "nth stage" for any n.