The conjugacy problem for UPG elements of $Out(F_n)$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2019
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| _version_ | 1866918077007921152 |
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| author | Feighn, Mark Handel, Michael |
| author_facet | Feighn, Mark Handel, Michael |
| contents | An element $ϕ$ of the outer automorphism group $\Out(\f)$ of the rank $n$ free group $F_n$ is {\it polynomially growing} if the word lengths of conjugacy classes in $\f$ grow at most polynomially under iteration by $ϕ$. It is {\it unipotent} if additionally its action on the first homology of $\f$ with integer coefficients is unipotent. In particular, if $ϕ$ is polynomially growing and acts trivially on first homology with coefficients the integers mod 3 then $ϕ$ is unipotent and also every polynomially growing element has a positive power that is unipotent. We solve the conjugacy problem in $\Out(\f)$ for the subset of unipotent elements. Specifically, there is an algorithm that decides if two such are conjugate in $\Out(\f)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1906_04147 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The conjugacy problem for UPG elements of $Out(F_n)$ Feighn, Mark Handel, Michael Group Theory 20F65 57M07 An element $ϕ$ of the outer automorphism group $\Out(\f)$ of the rank $n$ free group $F_n$ is {\it polynomially growing} if the word lengths of conjugacy classes in $\f$ grow at most polynomially under iteration by $ϕ$. It is {\it unipotent} if additionally its action on the first homology of $\f$ with integer coefficients is unipotent. In particular, if $ϕ$ is polynomially growing and acts trivially on first homology with coefficients the integers mod 3 then $ϕ$ is unipotent and also every polynomially growing element has a positive power that is unipotent. We solve the conjugacy problem in $\Out(\f)$ for the subset of unipotent elements. Specifically, there is an algorithm that decides if two such are conjugate in $\Out(\f)$. |
| title | The conjugacy problem for UPG elements of $Out(F_n)$ |
| topic | Group Theory 20F65 57M07 |
| url | https://arxiv.org/abs/1906.04147 |