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Register a new userCoarse $mathcal{Z}$-Boundaries for Groups
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We generalize Bestvinas notion of a $mathcal{Z}$-boundary for a group to that of a coarse $mathcal{Z}$-boundary. We show that established theorems about $mathcal{Z}$-boundaries carry over nicely to the more general theory, and that some wished-for properties of $mathcal{Z}$-boundaries become theorems when applied to coarse $mathcal{Z}$-boundaries. Most notably, the property of admitting a coarse $mathcal{Z}$-boundary is a pure quasi-isometry invariant. In the process, we streamline both new and existing definitions by introducing the notion of a model $mathcal{Z}$-geometry. In accordance with the existing theory, we also develop an equivariant version of the above -- that of a coarse $Emathcal{Z}$-boundary.
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In this work we introduce a new combinatorial notion of boundary $Re C$ of an $omega$-dimensional cubing $C$. $Re C$ is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of $C$, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When $C$ arises as the dual of a cubulation -- or discrete system of halfspaces -- $HH$ of a CAT(0) space $X$ (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how $HH$ induces a function $rho:bd XtoRe C$. We develop a notion of uniformness for $HH$, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair $(X,HH)$ admits a geometric action by a group $G$, then the fibers of $rho$ form a stratification of $bd X$ graded by the order structure of $Re C$. We also show how this structure computes the components of the Tits boundary of $X$. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of $C$, we give a condition for the co-compactness of the action of $G$ on $C$ in terms of $rho$, generalizing a result of Williams, previously known only for Coxeter groups.
A $mathcal{Z}$-structure on a group $G$ was introduced by Bestvina in order to extend the notion of a group boundary beyond the realm of CAT(0) and hyperbolic groups. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathcal{EZ}$-structure. The general questions of which groups admit $mathcal{Z}$- or $mathcal{EZ}$-structures remain open. In this paper we add to the current knowledge by showing that all Baumslag-Solitar groups admit $mathcal{EZ}$-structures and all generalized Baumslag-Solitar groups admit $mathcal{Z}$-structures.
Bestvina introduced a $mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $mathcal{E}mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $mathcal{E}mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $mathcal{E}mathcal{Z}$-structures on Baumslag-Solitar groups and $mathcal{Z}$-structures on generalized Baumslag-Solitar groups.
We develop a coarse notion of bundle and use it to understand the coarse geometry of group extensions and, more generally, groups acting on proper metric spaces. The results are particularly sharp for groups acting on (locally finite) trees with Abelian stabilizers, which we are able to classify completely.
We show that coarse property C is preserved by finite coarse direct products. We also show that the coarse analog of Dydaks countable asymptotic dimension is equivalent to the coarse version of straight finite decomposition complexity and is therefore preserved by direct products.
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