论文标题
Helly群体
Helly groups
论文作者
论文摘要
Helly图是每个成对相交球的家族都有非空交集的图。这是经典且研究的图表类。在本文中，我们着重于在Helly图上几何作用的组 - Helly组。我们提供了许多此类组的示例：所有（Gromov）双曲线，CAT（0）立方，有限的图形C（4）$ - $ t（4）小型取消组，以及在欧几里得建筑物$ C_N $的欧几里得建筑物中提供类型的均匀晶格；与有限亚组合并的Helly组的免费产品，Helly组的图形产品，Helly组的某些图产品，一些Helly组的右角图以及有限的正常亚组的Helly组的商。我们展示了Helly组的许多特性：双重性，存在有限维度模型，用于分类空间，以进行适当的动作，渐近锥的合同，EZ-goundaries的存在，Farrell-Jones猜想的满意度和粗糙的Baum-Connes猜想。这为某些古典群体（例如FC型Artin组）和统一的结果提供了新的结果。
Helly graphs are graphs in which every family of pairwise intersecting balls has a non-empty intersection. This is a classical and widely studied class of graphs. In this article we focus on groups acting geometrically on Helly graphs -- Helly groups. We provide numerous examples of such groups: all (Gromov) hyperbolic, CAT(0) cubical, finitely presented graphical C(4)$-$T(4) small cancellation groups, and type-preserving uniform lattices in Euclidean buildings of type $C_n$ are Helly; free products of Helly groups with amalgamation over finite subgroups, graph products of Helly groups, some diagram products of Helly groups, some right-angled graphs of Helly groups, and quotients of Helly groups by finite normal subgroups are Helly. We show many properties of Helly groups: biautomaticity, existence of finite dimensional models for classifying spaces for proper actions, contractibility of asymptotic cones, existence of EZ-boundaries, satisfiability of the Farrell-Jones conjecture and of the coarse Baum-Connes conjecture. This leads to new results for some classical families of groups (e.g. for FC-type Artin groups) and to a unified approach to results obtained earlier.
