Generalizing the Bieri-Neumann-Strebel-Renz Invariants, this Memoir presents the foundations of a theory of (not necessarily discrete) actions \(\rho\) of a (suitable) group \(G\) by isometries on a proper CAT(0) space \(M\). The passage from groups \(G\) to group actions \(\rho\) implies the introduction of “Sigma invariants” \(\Sigma^k(\rho)\) to replace the previous \(\Sigma^k(G)\) introduced by those authors. Their theory is now seen as a special case of what is studied here so that readers seeking a detailed treatment of their theory will find it included here as a special case.

We define and study “controlled \(k\)-connectedness \((CC^k)\)” of \(\rho\), both over \(M\) and over end points \(e\) in the “boundary at infinity” \(\partial M\); \(\Sigma^k(\rho)\) is by definition the set of all \(e\) over which the action is \((k-1)\)-connected. A central theorem, the Boundary Criterion, says that \(\Sigma^k(\rho) = \partial M\) if and only if \(\rho\) is \(CC^{k-1}\) over \(M\). An Openness Theorem says that \(CC^k\) over \(M\) is an open condition on the space of isometric actions \(\rho\) of \(G\) on \(M\). Another Openness Theorem says that \(\Sigma^k(\rho)\) is an open subset of \(\partial M\) with respect to the Tits metric topology. When \(\rho(G)\) is a discrete group of isometries the property \(CC^{k-1}\) is equivalent to ker\((\rho)\) having the topological finiteness property “type \(F_k\)”. More generally, if the orbits of the action are discrete, \(CC^{k-1}\) is equivalent to the point-stabilizers having type \(F_k\). In particular, for \(k=2\) we are characterizing finite presentability of kernels and stabilizers.

Examples discussed include: locally rigid actions, translation actions on vector spaces (especially those by metabelian groups), actions on trees (including those of \(S\)-arithmetic groups on Bruhat-Tits trees), and \(SL_2\) actions on the hyperbolic plane.