论文标题
右角建筑物中格子的可相当性
Commensurability of lattices in right-angled buildings
论文作者
论文摘要
令$γ$是有限组的图形产品，具有有限的基础图，让$δ$为相关的右角建筑物。我们证明，当立方体构成$ up $（δ）$中的统一晶格$λ$在$γ$的情况下弱弱，并且仅当所有$λ$的凸子组都是可分离的。作为推论，带有通用盖$δ$的任何两个有限的特殊立方体综合体都有一个共同的有限盖。我们定理的一个重要特殊案例是$γ$是一个直角的Coxeter组，而$δ$是相关的戴维斯综合体。我们还为右角Artin组获得了类似的结果。此外，当$δ$具有紫红色建筑物的结构时，我们推断出$γ$的准静电刚度。
Let $Γ$ be a graph product of finite groups, with finite underlying graph, and let $Δ$ be the associated right-angled building. We prove that a uniform lattice $Λ$ in the cubical automorphism group Aut$(Δ)$ is weakly commensurable to $Γ$ if and only if all convex subgroups of $Λ$ are separable. As a corollary, any two finite special cube complexes with universal cover $Δ$ have a common finite cover. An important special case of our theorem is where $Γ$ is a right-angled Coxeter group and $Δ$ is the associated Davis complex. We also obtain an analogous result for right-angled Artin groups. In addition, we deduce quasi-isometric rigidity for the group $Γ$ when $Δ$ has the structure of a Fuchsian building.
