Groups with no coarse embeddings into hyperbolic groups
Groups with no coarse embeddings into hyperbolic groups
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发表时间：
2017-02
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通讯作者：
David Hume;A. Sisto
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作者：
David Hume;A. Sisto
我们引入了一个障碍，以阻止给定群或空间粗嵌入到双曲群中，或者更一般地，嵌入到有界度的双曲图中。我们考虑的条件是“指数地接纳许多胖bigons”，并且它通过在有界度的图之间的粗嵌入来保持。具有指数增长和线性发散的群（例如两个群的直积，其中一个群具有指数增长，可解群不是虚幂零的，以及均匀高阶格）具有此性质，而双曲图则没有，因此前者不能粗略地嵌入后者。其他的例子包括某些缺双曲群和某些小消去群。
We introduce an obstruction to the existence of a coarse embedding of a given group or space into a hyperbolic group, or more generally into a hyperbolic graph of bounded degree. The condition we consider is "admitting exponentially many fat bigons", and it is preserved by a coarse embedding between graphs with bounded degree. Groups with exponential growth and linear divergence (such as direct products of two groups one of which has exponential growth, solvable groups that are not virtually nilpotent, and uniform higher-rank lattices) have this property and hyperbolic graphs do not, so the former cannot be coarsely embedded into the latter. Other examples include certain lacunary hyperbolic and certain small cancellation groups.