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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
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 Click above image for expanded view 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.

• 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.

• 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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