图上Z^2可扩作用的非存在性

The Non-Existence of Expansive Z^2 Actions on Graphs

文[9]中作者考虑连续统上可扩群作用的存在性问题,证明了单位闭区间上不存在自由交换群Z×Z的可扩作用,并且给出一个例子表明闭区间上存在自由积Z*Z的可扩作用.换句话说,由两个交换同胚生成的群是不能可扩作用在闭区间上的,但还是存在由两个非交换同胚生成的群能够可扩作用在闭区间上.本文证明了图上不存在Z×Z的可扩作用,解决了文[9]所提的一个问题.

In [9], we considered the problem of whether a continuum admits expansive group actions, and proved that the free Abelian group Z ×Z can not act expansively on the unit intenval I; an example of expansive Z * Z action on was also given where Z * Z denotes the free product of two copies of Z. That is to say, though the group generated by two commutative homeomorphisms can not act on an arc expansively, there still exists an expansive group action which is generated by two noncommutative homeomorphisms. In this paper, we prove that there exists no expansive Z2 actions on graphs, which answers one of the quenstions in paper [9].