论文标题
$ \ mathrm {out}（f_n）$的Abelian子组的自由组和中心化的外部自动形态的根
Roots of outer automorphisms of free groups and centralizers of abelian subgroups of $\mathrm{Out}(F_N)$
论文作者
论文摘要
令$ n \ geq 2 $，然后让$ \ mathrm {out}（f_n）$为nonabelian自由组的外部自动形态组，等级$ n $。令$ \ mathrm {ia} _n（\ mathbb {z}/3 \ mathbb {z}）$为$ \ mathrm {out}（out}（f_n）$的有限索引子组，这是$ \ mathrm {out}（out}（f_n）$的自然动作的内核$ h_1（f_n，\ mathbb {z}/3 \ mathbb {z}）$。我们表明，$ \ mathrm {ia} _n（\ Mathbb {z}/3 \ Mathbb {z}）$是$ r $ -group，也就是说，对于每个$ ϕ，ψ\ in \ in \ in \ in \ mathrm {ia} _n（ia} _n（ia} _n（\ mathbb）， \ mathbb {n}^*$，这样$ ϕ^k =ψ^k $，然后$ ϕ =ψ$。这回答了Handel和Mosher的问题。然后，我们使用以下事实：$ \ mathrm {ia} _n（\ Mathbb {z}/3 \ Mathbb {z}）$是$ r $ - 组，以证明$ \ Mathrm {ia} _n（ia} _n（\ mathbb {z}/3 \ althbb {z}/3 \ mathb}的归一化量$ \ mathrm {ia} _n（\ mathbb {z}/3 \ mathbb {z}）$等于其centurnizer。由于Feighn和Handel，我们最终给出了结果的替代证明，即$ \ mathrm {out}（f_n）$的元素的中心设备只有许多周期性的循环类别的最大循环子组的$ f_n $的最大循环子组实际上是Abelian。
Let $N \geq 2$ and let $\mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ be the finite index subgroup of $\mathrm{Out}(F_N)$ which is the kernel of the natural action of $\mathrm{Out}(F_N)$ on $H_1(F_N,\mathbb{Z}/3\mathbb{Z})$. We show that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group, that is, for every $ϕ,ψ\in \mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$, if there exists $k \in \mathbb{N}^*$ such that $ϕ^k=ψ^k$, then $ϕ=ψ$. This answers a question of Handel and Mosher. We then use the fact that $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is an $R$-group in order to prove that the normalizer in $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ of every abelian subgroup of $\mathrm{IA}_N(\mathbb{Z}/3\mathbb{Z})$ is equal to its centralizer. We finally give an alternative proof of a result, due to Feighn and Handel, that the centralizer of an element of $\mathrm{Out}(F_N)$ which has only finitely many periodic orbits of conjugacy classes of maximal cyclic subgroups of $F_N$ is virtually abelian.
