Memoirs of the American Mathematical Society 2003; 83 pp; softcover Volume: 161 ISBN10: 0821831844 ISBN13: 9780821831847 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/161/765
 Generalizing the BieriNeumannStrebelRenz 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 \((k1)\)connected. A central theorem, the Boundary Criterion, says that \(\Sigma^k(\rho) = \partial M\) if and only if \(\rho\) is \(CC^{k1}\) 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 \(CC^{k1}\) is equivalent to ker\((\rho)\) having the topological finiteness property "type \(F_k\)". More generally, if the orbits of the action are discrete, \(CC^{k1}\) is equivalent to the pointstabilizers 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 BruhatTits trees), and \(SL_2\) actions on the hyperbolic plane. Readership Graduate student and research mathematicians. Table of Contents Part 1. Controlled Connectivity and Openness Results  Outline, main results and examples
 Technicalities concerning the \(CC^{n1}\) property
 Finitary maps and sheaves of maps
 Sheaves and finitary maps over a control space
 Construction of sheaves with positive shift
 Controlled connectivity as an open condition
 Completion of the proofs of Theorems A and A'
 The invariance theorem
Part 2. The Geometric Invariants  Short summary of Part 2
 Outline, main results and examples
 Further technicalities on \(\mathrm{CAT}(0)\) spaces
 \(CC^{n1}\) over endpoints
 Finitary contractions towards endpoints
 From \(CC^{n1}\) over endpoints to contractions
 Proofs of Theorems EH
 Appendix A: Alternative formulations of \(CC^{n1}\)
 Appendix B: Further formulations of \(CC^{n1}\)
 Bibliography
