arXiv 1306.56882013-06-24math.GR

自由群的树不可约自同构

Tree-irreducible automorphisms of free groups

Martin Lustig

摘要：

我们引入一类新的自同构$φ$的非阿贝尔自由群$F_N$有限秩$N \geq 2$，其中包含所有iwips（=完全不可约自同构），但也任何自同构诱导的伪Anosov同胚的表面与任意多个边界组件。更一般地，可以存在秩为$\geq 2$的$F_N$的子组，其中$φ$限制于身份。我们证明了一些基本的事实，这样的{ \em树不可约}自同构，并表明，连同Dehn扭曲自同构，他们是自然的基本积木，任何自同构的$\FN$可以构建在一个训练轨道设置。然后，我们显示：{ \bf定理：} { \it $F_N$的每一个树不可约自同构都在外层空间的Thurston紧化$\bar{\rm CV}_N$上诱导了南北动力学。}最后，我们定义了一个“爆破”结构的顶点上的训练轨道地图，其中，从iwip开始，产生树不可约的自同构，在一般情况下不是iwip。

英文摘要：

We introduce a new class of automorphisms $φ$ of the non-abelian free group $F_N$ of finite rank $N \geq 2$ which contains all iwips (= fully irreducible automorphisms), but also any automorphism induced by a pseudo-Anosov homeomorphism of a surface with arbitrary many boundary components. More generally, there may be subgroups of $F_N$ of rank $\geq 2$ on which $φ$ restricts to the identity. We prove some basic facts about such {\em tree-irreducible} automorphisms, and show that, together with Dehn twist automorphisms, they are the natural basic building blocks from which any automorphism of $\FN$ can be constructed in a train track set-up. We then show: {\bf Theorem:} {\it Every tree-irreducible automorphism of $F_N$ has induced North-South dynamics on the Thurston compactification $\bar{\rm CV}_N$ of Outer space.} Finally, we define a "blow-up" construction on the vertices of a train track map, which, starting from iwips, produces tree-irreducible automorphisms which in general are not iwip.

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