Connectivity properties of group actions on non-positively curved spaces
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Bibliographic Information
Connectivity properties of group actions on non-positively curved spaces
（Memoirs of the American Mathematical Society, no. 765）
American Mathematical Society, 2003
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Includes bibliographical references (p. 81-83)
"January 2003, volume 161, number 765 (second of 5 numbers)"
Description and Table of Contents
Description
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
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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