2017/12/15 by Robert Bieri, Bieri, Robert, Ross Geoghegan +1
Mathematics · #14T05 #20E42 #20F65 #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1712.05480
openalex publication_date 2017/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.