局部有限树上作用的连通性
Connectivity Properties for Actions on Locally Finite Trees
摘要：
给定一个局部有限树T上有限生成群G的作用，我们将可视边界\partialT上的点看作T中的方向，并使用ρ将这个方向提升到G。对于每个点E \in \partialT，这允许我们询问G是否在“E的方向”上是（n - 1）-连通的。不变量^n（ρ）\subseteq \partialT则记录G是（n-1）-连通的方向集。本文引入了一类作用，通过分析树间的商映射，可以计算出它们的π ^1（ρ）.我们证明，对于这类作用，在合理的假设下，ρ 1（ρ）只由一个点组成。通过加强假设，我们能够精确地描述给定的端点何时位于对任何n的^n（ρ）中。
英文摘要：
Given an action by a finitely generated group G on a locally finite tree T, we view points of the visual boundary \partialT as directions in T and use ρ to lift this sense of direction to G. For each point E \in \partialT, this allows us to ask if G is (n - 1)-connected "in the direction of E". The invariant Σ^n(ρ) \subseteq \partialT then records the set of directions in which G is (n-1)-connected. In this paper, we introduce a family of actions for which Σ^1(ρ) can be calculated through analysis of certain quotient maps between trees. We show that for actions of this sort, under reasonable hypotheses, Σ1(ρ) consists of no more than a single point. By strengthening the hypotheses, we are able to characterize precisely when a given end point lies in Σ^n(ρ) for any n.
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