Helly群、粗Helly群与相对双曲性
Helly groups, coarsely Helly groups, and relative hyperbolicity
AI总结：
本文证明相对于Helly或粗Helly子群相对双曲的有限生成群仍为Helly或粗Helly群，并刻画抛物及拟凸子群的相应性质。
摘要：
如果一个单纯图中任意两两相交的球族都具有非空的（粗）交，则称该图为（粗）Helly图。（粗）Helly群是指在（粗）Helly图上进行几何作用的群。我们的主要结果是：相对于（粗）Helly子群呈相对双曲的有限生成群，其本身也是（粗）Helly群。一个重要推论是，包括环面相对双曲群在内的各种经典群都具有类似CAT(0)的结构——它们在带有凸测地双梳的空间上进行几何作用。作为证明主要定理的手段，我们建立了一个关于相对双曲群且具有独立意义的结果：对相对双曲群在其上进行几何作用的图中的测地线，给出一种“相对双曲”描述。在反方向上，我们证明对于相对双曲的（粗）Helly群，其抛物子群也是（粗）Helly群。更一般地，我们证明（粗）Helly群的“拟凸”子群本身也是（粗）Helly群。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
