Similar Relatively Hyperbolic Actions of a Group
Similar Relatively Hyperbolic Actions of a Group
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DOI：
10.1093/imrn/rnv170
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发表时间：
2013-05
影响因子：
1
通讯作者：
V. Gerasimov;L. Potyagailo
中科院分区：
文献类型：
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作者：
V. Gerasimov;L. Potyagailo
设离散群G在紧向量X和紧向量Y上具有两个同胚收敛作用。考虑以下问题：在紧致Z和连续等变映射X←Z→Y上是否存在收敛作用GyZ ？我们称空间Z（以及G对其的作用）为回拉空间（作用）。在这种大背景下，O. Baker和T. Riley最近的研究结果给出了否定的答案[BR]。另外，假设初始作用是相对双曲的，也就是说，它们是非抛物线的，不同对上的诱导作用是紧致的。则由[Ge2]可知G有限生成时的回拉空间的存在性。本文的主要结果表明，当且仅当其中一个作用的极大抛物子群是另一个作用的动态拟凸时，存在回拉空间。我们给出了可数秩自由群G的两个不存在回拉作用的相对双曲作用的例子。我们研究了相对双曲群的测地线流概念的类比。进一步用这些结果证明了主要定理。
Let a discrete group G possess two convergence actions by homeomorphisms on compacta X and Y . Consider the following question: does there exist a convergence action GyZ on a compactum Z and continuous equivariant maps X ← Z → Y ? We call the space Z (and action of G on it) pullback space (action). In such general setting a negative answer follows from a recent result of O. Baker and T. Riley [BR]. Suppose, in addition, that the initial actions are relatively hyperbolic that is they are non-parabolic and the induced action on the distinct pairs are cocompact. Then the existence of the pullback space if G is finitely generated follows from [Ge2]. The main result of the paper claims that the pullback space exists if and only if the maximal parabolic subgroups of one of the actions are dynamically quasiconvex for the other one. We provide an example of two relatively hyperbolic actions of the free group G of countable rank for which the pullback action does not exist. We study an analog of the notion of geodesic flow for relatively hyperbolic groups. Further these results are used to prove the main theorem.