Connectivity Properties of Group Actions on Non-Positively Curved Spaces PDF
by Robert Bieri
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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.
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Download Now
- Format:PDF
- Pages:83 pages
- Publisher:American Mathematical Society
- Publication Date:01/01/1900
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- ISBN:9781470403638
Information
-
Download Now
- Format:PDF
- Pages:83 pages
- Publisher:American Mathematical Society
- Publication Date:01/01/1900
- Category:
- ISBN:9781470403638