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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

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$60
Individual Members: US$36
Institutional Members: US$48
Order Code: MEMO/161/765
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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.


Graduate student and research mathematicians.

Table of Contents

  • 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
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