论文标题
Helly组，粗helly组和相对双曲线
Helly groups, coarsely Helly groups, and relative hyperbolicity
论文作者
论文摘要
如果任何成对相交的球具有非空（粗）相交，则据说（粗糙）据称是（粗糙的）Helly。 （粗糙的）Helly组是在（粗糙的）Helly图上几何作用的组。我们的主要结果是，相对于（粗糙的）Helly亚组的有限生成的组是（粗糙的）Helly。一个重要的结果是，包括托拉尔相对双曲线在内的各种经典群体都配备了猫（0）类似结构 - 它们在具有凸deevex geodesic dicombing的空间上作用。作为证明主要定理的一种手段，我们建立了有关相对双曲群的独立兴趣的结果：在图中，对地球学的“相对双曲”描述，相对双曲线群体在几何上作用着几何形式。在另一个方向上，我们表明，对于相对双曲线（粗糙）的helly组，它们的抛物线亚组也（粗糙）也是Helly。更普遍地，我们表明（粗糙的）Helly群体的“ Quasiconvex”子组本身（粗糙）Helly。
A simplicial graph is said to be (coarsely) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. (Coarsely) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is that finitely generated groups that are hyperbolic relative to (coarsely) Helly subgroups are themselves (coarsely) Helly. One important consequence is that various classical groups, including toral relatively hyperbolic groups, are equipped with a CAT(0)-like structure -- they act geometrically on spaces with convex geodesic bicombing. As a means of proving the main theorems we establish a result of independent interest concerning relatively hyperbolic groups: a `relatively hyperbolic' description of geodesics in a graph on which a relatively hyperbolic group acts geometrically. In the other direction, we show that for relatively hyperbolic (coarsely) Helly groups their parabolic subgroups are (coarsely) Helly as well. More generally, we show that `quasiconvex' subgroups of (coarsely) Helly groups are themselves (coarsely) Helly.
