论文标题
$ \ text {out} \ left（f_ {n} \ right）$的表示形式，来计数循环单词的子字
A representation of $\text{Out}\left(F_{n}\right)$ by counting subwords of cyclic words
论文作者
论文摘要
我们将Rapaport和Higgins的组合方法概括为Whitehead算法。我们表明，对于每一个免费组$ f $的自动形态$φ$ $ f $和每个单词$ u \ in f $中都存在有限的单词$ s_ {u，φ} $满足以下属性的单词：对于每个环状单词$ w $，$ w $，$ u $ u $ unting of subnord of sub of usportance un（weft）in Int of（w pright）的数量， $ s_ {u，φ} $作为$ W $的子字。我们使用这个事实来构建$ \ text {out} \ left（f_ {n} \右）$的忠实表示，在$ \ mathbb {z} $ - 模块的反度上，每个自动形态都可以通过有限的矩形矩阵来代表，这可以将其视为相结合的近似近似近似，从而可以将其视为近似近似的近似值。
We generalize the combinatorial approaches of Rapaport and Higgins--Lyndon to the Whitehead algorithm. We show that for every automorphism $φ$ of a free group $F$ and every word $u\in F$ there exists a finite multiset of words $S_{u,φ}$ satisfying the following property: For every cyclic word $w$, the number of times $u$ appears as a subword of $φ\left(w\right)$ depends only on the appearances of words in $S_{u,φ}$ as subwords of $w$. We use this fact to construct a faithful representation of $\text{Out}\left(F_{n}\right)$ on an inverse limit of $\mathbb{Z}$-modules, so that each automorphism is represented by sequence of finite rectangular matrices, which can be seen as successively better approximations of the automorphism.
