Connectivity Properties of Group Actions on Non-Positively Curved Spaces

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 $M$. It introduces the passage from groups $G$ to group actions $\rho$ that implies the introduction of 'Sigma invariants'.