Morse拟测地线的一个量化的局部到整体原理
A quantified local-to-global principle for Morse quasigeodesics
摘要：
在arXiv：1403.7671中，Kapovich、Leeb和Porti给出了Anosov表示$Γ\to G$的几个新特征，其中包括双曲群$Γ$中的测地线映射到相关对称空间$G/K$中的“Morse拟测地线”。与负曲率设置类似，他们证明了Morse拟测地线的局部到全局原理，并描述了一种算法，该算法可以在有限时间内验证给定表示的Anosov性质。然而，他们的证明的某些部分涉及非构造性的紧凑性和限制性参数，因此他们的定理并没有明确量化需要检查以保证全局Morse行为的局部邻域的大小。在本文中，我们补充了他们的工作与对称空间中的估计，以获得他们的局部到整体原则的第一个明确的标准。这使得他们验证Anosov性质的算法有效。作为应用，我们演示了如何计算显式扰动邻域的Anosov表示与两个例子。
英文摘要：
In arXiv:1403.7671, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations $Γ\to G$, including one where geodesics in the word hyperbolic group $Γ$ map to "Morse quasigeodesics" in the associated symmetric space $G/K$. In analogy with the negative curvature setting, they prove a local-to-global principle for Morse quasigeodesics and describe an algorithm which can verify the Anosov property of a given representation in finite time. However, some parts of their proof involve non-constructive compactness and limiting arguments, so their theorem does not explicitly quantify the size of the local neighborhoods one needs to examine to guarantee global Morse behavior. In this paper, we supplement their work with estimates in the symmetric space to obtain the first explicit criteria for their local-to-global principle. This makes their algorithm for verifying the Anosov property effective. As an application, we demonstrate how to compute explicit perturbation neighborhoods of Anosov representations with two examples.
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