  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.
