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Connectivity Properties of Group Actions on Non-Positively Curved Spaces
 
Robert Bieri University of Frankfurt, Frankfurt, Germany
Ross Geoghegan Binghamton University, Binghamton, NY
Front Cover for Connectivity Properties of Group Actions on Non-Positively Curved Spaces
Available Formats:
Electronic ISBN: 978-1-4704-0363-8
Product Code: MEMO/161/765.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
Front Cover for Connectivity Properties of Group Actions on Non-Positively Curved Spaces
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  • Front Cover for Connectivity Properties of Group Actions on Non-Positively Curved Spaces
  • Back Cover for Connectivity Properties of Group Actions on Non-Positively Curved Spaces
Connectivity Properties of Group Actions on Non-Positively Curved Spaces
Robert Bieri University of Frankfurt, Frankfurt, Germany
Ross Geoghegan Binghamton University, Binghamton, NY
Available Formats:
Electronic ISBN:  978-1-4704-0363-8
Product Code:  MEMO/161/765.E
List Price: $60.00
MAA Member Price: $54.00
AMS Member Price: $36.00
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1612003; 83 pp
    MSC: Primary 20; 53; 57;

    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.

    Readership

    Graduate 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 $CC^{n-1}$ 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. $CC^{n-1}$ over endpoints
    • 13. Finitary contractions towards endpoints
    • 14. From $CC^{n-1}$ over endpoints to contractions
    • 15. Proofs of Theorems E-H
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Volume: 1612003; 83 pp
MSC: Primary 20; 53; 57;

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.

Readership

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 $CC^{n-1}$ 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. $CC^{n-1}$ over endpoints
  • 13. Finitary contractions towards endpoints
  • 14. From $CC^{n-1}$ over endpoints to contractions
  • 15. Proofs of Theorems E-H
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