eBook ISBN:  9781470403638 
Product Code:  MEMO/161/765.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 
eBook ISBN:  9781470403638 
Product Code:  MEMO/161/765.E 
List Price:  $60.00 
MAA Member Price:  $54.00 
AMS Member Price:  $36.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 161; 2003; 83 ppMSC: Primary 20; 53; 57
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.
ReadershipGraduate student and research mathematicians.

Table of Contents

Chapters

1. Introduction

Part 1. Controlled connectivity and openness results

2. Outline, main results and examples

3. Technicalities concerning the $CC^{n1}$ property

4. Finitary maps and sheaves of maps

5. Sheaves and finitary maps over a control space

6. Construction of sheaves with positive shift

7. Controlled connectivity as an open condition

8. Completion of the proofs of Theorems A and A′

9. The invariance theorem

Part 2. The geometric invariants

10. Outline, main results and examples

11. Further technicalities on CAT(0) spaces

12. $CC^{n1}$ over endpoints

13. Finitary contractions towards endpoints

14. From $CC^{n1}$ over endpoints to contractions

15. Proofs of Theorems EH


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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.
Graduate student and research mathematicians.

Chapters

1. Introduction

Part 1. Controlled connectivity and openness results

2. Outline, main results and examples

3. Technicalities concerning the $CC^{n1}$ property

4. Finitary maps and sheaves of maps

5. Sheaves and finitary maps over a control space

6. Construction of sheaves with positive shift

7. Controlled connectivity as an open condition

8. Completion of the proofs of Theorems A and A′

9. The invariance theorem

Part 2. The geometric invariants

10. Outline, main results and examples

11. Further technicalities on CAT(0) spaces

12. $CC^{n1}$ over endpoints

13. Finitary contractions towards endpoints

14. From $CC^{n1}$ over endpoints to contractions

15. Proofs of Theorems EH