$Out(F_r)$上的训练轨道定向随机游动
A train track directed random walk on $Out(F_r)$
摘要：
Rivin、Calegari-Maher和Sisto的几个已知结果表明，在$Out(F_r)$上的简单随机游动的$n$步骤之后获得的元素$ϕ_n\in Out(F_r)$是完全不可约的，概率趋于1，如$n\to\infty$。本文在$Out(F_r)$（其中$r\ge 3$）上构造了一个自然的“火车轨道定向”随机游动$\mathcal W$.我们证明，对于元$ϕ_n\in Out(F_r)$，在此随机游动的$n$步骤之后获得，具有渐近正概率的元$ϕ_n$具有以下性质：$ϕ_n$是一个年龄完全不可约的，它允许一个没有周期性尼尔森路径的火车轨道代表和一个非退化的非法转弯，$ϕ_n$具有“无旋转索引”$\frac{3}{2}-r$（因此$ϕ_n$的吸引树$T_{ϕ_n}$的几何索引是$2r-3$），有索引表$\{\frac{3}{2}-r\}$，理想Whitehead图是$2r-1$顶点上的完全图，外层空间$CV_r$中$ϕ_n$的轴丛由单轴组成.
英文摘要：
Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $ϕ_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In this paper we construct a natural "train-track directed" random walk $\mathcal W$ on $Out(F_r)$ (where $r\ge 3$). We show that, for the element $ϕ_n\in Out(F_r)$, obtained after $n$ steps of this random walk, with asymptotically positive probability the element $ϕ_n$ has the following properties: $ϕ_n$ is an ageometric fully irreducible, which admits a train-track representative with no periodic Nielsen paths and exactly one nondegenerate illegal turn, that $ϕ_n$ has "rotationless index" $\frac{3}{2}-r$ (so that the geometric index of the attracting tree $T_{ϕ_n}$ of $ϕ_n$ is $2r-3$), has index list $\{\frac{3}{2}-r\}$ and the ideal Whitehead graph being the complete graph on $2r-1$ vertices, and that the axis bundle of $ϕ_n$ in the Outer space $CV_r$ consists of a single axis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
