论文标题
雪花组的渐近锥和强捷径
Asymptotic cones of snowflake groups and the strong shortcut property
论文作者
论文摘要
我们展示了一个无限的雪花群家族，其渐近锥完全连接起来。我们的群体既没有多项式生长，也没有二次dehn功能，这是该现象的两个常见来源。我们进一步表明，我们每个组都有一个渐近锥，该锥体包含一个异小嵌入的圆圈，或者等效地，具有不强烈快捷方式的Cayley图。这些是渐近锥体包含“指标非平凡”环但没有拓扑非平凡的组的第一个示例。
We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected. Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon. We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut. These are the first examples of groups whose asymptotic cones contain `metrically nontrivial' loops but no topologically nontrivial ones.
