Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

S. Francaviglia, A. Martino
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引用次数: 9

Abstract

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.
自由群I自同构的位移:位移函数、最小点和火车轨道
这是我们研究Culler-Vogtmann外空间上自由群(更一般地说,自由积)的自同构位移函数的性质及其关于Lipschitz度规的简单化——自由分裂复形的两篇论文中的第一篇。不可约自同构的理论已经得到了很好的发展,我们集中讨论可约的情况。由于我们处理的是边界化,所以我们在更一般的变形空间设置中开发了所有需要的工具,以及它们相关的自由分裂复合体。本文研究了位移函数的局部性质。特别地,我们研究了它的凸性和在边界点上的行为,通过几何表征它的连续点。我们证明了$Aut(F_n)$的全局简单位移谱是$\mathbb R$的良序子集,这对算法的目的很有帮助。我们引入了列车轨道的一个较弱的概念,我们称之为{\em偏列车轨道}(它与不可约自同构的通常概念一致),并且我们证明,对于任何自同构,最小位移点- minpoints -与支持偏列车轨道的标记度量图一致。我们证明了任何自同构,无论是否可约,在外空间或其边界上都有一个部分火车轨道(即最小点)。我们证明,给定一个自同构,它的任意不变自由因子在部分列车轨道图中都是可见的。在后续的论文中,我们将证明位移函数的水平集是连通的,并将该结果应用于解决某些决策问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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