# Connectivity Properties of Group Actions on Non-Positively Curved Spaces

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*Robert Bieri; Ross Geoghegan*

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.

#### Table of Contents

# Table of Contents

## Connectivity Properties of Group Actions on Non-Positively Curved Spaces

- Contents vii8 free
- Preface xi12 free
- Chapter 1. Introduction 116 free
- Part 1. Controlled connectivity and openness results 520
- Chapter 2. Outline, Main Results and Examples 722
- 2.1. Non-positively curved spaces 722
- 2.2. Controlled connectivity: the definition of CC[sup(n-1)] 722
- 2.3. The case of discrete orbits 823
- 2.4. The Openness Theorem 924
- 2.5. Connections with Lie groups and local rigidity 1025
- 2.6. The new tool 1025
- 2.7. Summary of the core idea 1126
- 2.8. SL[sup(2)] examples 1126

- Chapter 3. Technicalities Concerning the CC[sup(n-1)]Property 1328
- Chapter 4. Finitary Maps and Sheaves of Maps 1732
- Chapter 5. Sheaves and Finitary Maps Over a Control Space 2338
- Chapter 6. Construction of Sheaves with Positive Shift 2944
- Chapter 7. Controlled Connectivity as an Open Condition 3550
- Chapter 8. Completion of the proofs of Theorems A and A' 4156
- Chapter 9. The Invariance Theorem 4358

- Part 2. The geometric invariants 4560
- Short summary of Part 2 4762
- Chapter 10. Outline, Main Results and Examples 4964
- Chapter 11. Further Technicalities on CAT(0) spaces 5974
- Chapter 12. CC[sup(n-1)] over Endpoints 6176
- Chapter 13. Finitary Contractions Towards Endpoints 6378
- Chapter 14. From CC[sup(n-1)] over Endpoints to Contractions 6782
- Chapter 15. Proofs of Theorems E-H 7186
- Appendix A: Alternative formulations of CC[sup(n-1)] 7590
- Appendix B: Further formulations of CC[sup(n-1)] 7792

- Bibliography 8196