论文标题
固定点索引和固定单词在图形的自我图中
Fixed point indices and fixed words at infinity of selfmaps of graphs
论文作者
论文摘要
固定点类的索引在尼尔森固定点理论中起着核心作用。 Jiang-Wang-Zhang证明，对于图和表面的自画，任何固定点类的索引都具有上限，称为其特征。 在本文中，我们研究了图形自图的索引和特征之间的差异。首先，对于自由组，我们将无限属性的固定单词扩展到了注射性内态。然后，通过使用相对火车轨道技术，我们表明上述差异很可能是吸引基本组引起的内态固定单词的等效类别的数量。由于两个吸引固定的单词和存在的特征都是由内态性本身决定的，因此我们给出了一种新的代数方法来估计图形自我图的固定点类别的索引。 结果，我们获得了一个上限，用于吸引自由群体的固定单词的固定单词，从而概括了由于Gaboriau-jaeger-levitt-lustig而导致的自动形态。此外，我们给出了一种简单的方法来粗略检测固定单词是否存在。
Indices of fixed point classes play a central role in Nielsen fixed point theory. Jiang-Wang-Zhang proved that for selfmaps of graphs and surfaces, the index of any fixed point class has an upper bound called its characteristic. In this paper, we study the difference between the index and the characteristic for selfmaps of graphs. First, for free groups, we extend attracting fixed words at infinity of automorphisms into that of injective endomorphisms. Then, by using relative train track technique, we show that the difference mentioned above is quite likely to be the number of equivalence classes of attracting fixed words of the endomorphism induced on the fundamental group. Since both of attracting fixed words and the existed characteristic are totally determined by endomorphisms themselves, we give a new algebraic approach to estimate indices of fixed point classes of graph selfmaps. As consequence, we obtain an upper bound for attracting fixed words of injective endomorphisms of free groups, generalizing the one for automorphisms due to Gaboriau-Jaeger-Levitt-Lustig. Furthermore, we give a simple approach to roughly detecting whether fixed words exist or not.
