The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X is locally finite, essential, irreducible, non-Euclidean and admits a cocompact action by a group G, Nevo—Sageev show that a subset, B(X), of the Roller boundary is the realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. Additionally, these authors show B(X) satisfies many other desirable dynamical and topological properties. In this article we give several equivalent characterizations for when B(X) is equal to the entire Roller boundary. As an application we show, under mild hypotheses, that if X is also 2-dimensional then X is G-equivariantly quasi-isometric to a CAT(0) cube complex X′ whose Roller boundary is equal to B(X′). Additionally, we use our characterization to show that the usual CAT(0) cube complex for which an infinite right-angled Coxeter/Artin group acts on geometrically has Roller boundary equal to B(X), as long as the corresponding group does not decompose as a direct product.

Roller边界是CAT（0）多维数据集X的众所周知的压缩。当X是局部有限的，基本的，不可约的，非欧几里得的并且接受G组的协紧作用时，Nevo-Sageev表明，Roller边界的一个子集B（X）是泊松边界的实现，并且的动作ģ上乙（X）是最小的，并且强烈的近端。此外，这些作者还证明B（X）满足许多其他所需的动力学和拓扑特性。在本文中，我们给出了B（X）等于整个Roller边界。作为一个应用，我们证明了在柔和的假设下，如果X也是二维的，则X是与Roller边界等于B（X '）的CAT（0）立方复合物X '相等的G-拟等距的。此外，我们使用特征描述了一个无限的直角Coxeter / Artin组在几何上起作用的常规CAT（0）立方体复合体，其Roller边界等于B（X），只要相应的组不分解即可。作为直接产品。