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- W4311726052 abstract "Let $N geq 2$ and let $mathrm{Out}(F_N)$ be the outer automorphism group of a nonabelian free group of rank $N$. Let $mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$ be the finite index subgroup of $mathrm{Out}(F_N)$ which is the kernel of the natural action of $mathrm{Out}(F_N)$ on $H_1(F_N,mathbb{Z}/3mathbb{Z})$. We show that $mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$ is an $R$-group, that is, for every $phi,psi in mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$, if there exists $k in mathbb{N}^*$ such that $phi^k=psi^k$, then $phi=psi$. This answers a question of Handel and Mosher. We then use the fact that $mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$ is an $R$-group in order to prove that the normalizer in $mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$ of every abelian subgroup of $mathrm{IA}_N(mathbb{Z}/3mathbb{Z})$ 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." @default.
- W4311726052 created "2022-12-28" @default.
- W4311726052 creator A5069653364 @default.
- W4311726052 date "2022-12-15" @default.
- W4311726052 modified "2023-09-25" @default.
- W4311726052 title "Roots of outer automorphisms of free groups and centralizers of abelian subgroups of $mathrm{Out}(F_N)$" @default.
- W4311726052 doi "https://doi.org/10.48550/arxiv.2212.07674" @default.
- W4311726052 hasPublicationYear "2022" @default.
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