No Arabic abstract
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.
We study the general theory of asymptotically CAT(0) groups, explaining why such a group has finitely many conjugacy classes of finite subgroups, is $F_infty$ and has solvable word problem. We provide techniques to combine asymptotically CAT(0) groups via direct products, amalgams and HNN extensions. The universal cover of the Lie group $PSL(2,mathbb{R})$ is shown to be an asymptotically CAT(0) metric space. Therefore, co-compact lattices in $widetilde{PSL(2,mathbb{R})}$ provide the first examples of asymptotically CAT(0) groups which are neither CAT(0) nor hyperbolic. Another source of examples is shown to be the class of relatively hyperbolic groups.
We show that any split extension of a right-angled Coxeter group $W_{Gamma}$ by a generating automorphism of finite order acts faithfully and geometrically on a $mathrm{CAT}(0)$ metric space.
We discuss a problem posed by Gersten: Is every automatic group which does not contain Z+Z subgroup, hyperbolic? To study this question, we define the notion of n-tracks of length n, which is a structure like Z+Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is weakly special, then the above question is answered affirmatively.
We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris-Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous.
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.