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Connectivity Properties of Group Actions on Non-Positively Curved Spaces
Robert Bieri, University of Frankfurt, Germany, and Ross Geoghegan, Binghamton University, NY
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Memoirs of the American Mathematical Society
2003; 83 pp; softcover
Volume: 161
ISBN-10: 0-8218-3184-4
ISBN-13: 978-0-8218-3184-7
List Price: US$57 Individual Members: US$34.20
Institutional Members: US\$45.60
Order Code: MEMO/161/765

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.

• Introduction
Part 1. Controlled Connectivity and Openness Results
• Outline, main results and examples
• Technicalities concerning the $$CC^{n-1}$$ 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
• $$CC^{n-1}$$ over endpoints
• Finitary contractions towards endpoints
• From $$CC^{n-1}$$ over endpoints to contractions
• Proofs of Theorems E-H
• Appendix A: Alternative formulations of $$CC^{n-1}$$
• Appendix B: Further formulations of $$CC^{n-1}$$
• Bibliography