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The conjugacy problem for polynomially growing elements of $Out(F_{n})$
Post-edited
Authors : Feighn, Mark (Author of the conference)
CIRM (Publisher )
20F65 - Geometric group theory
57M07 - Topological methods in group theory
CIRM (Publisher )
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| Conjugacy problem for Out (Fn) Eigengraphs and invariants Whitehead's theorem and reduction Algebraic invariants Induction Understanding rays |
Abstract :
(joint work with Michael Handel) $Out(F_{n}) := Aut(F_{n})/Inn(F_{n})$ denotes the outer automorphism group of the rank n free group $F_{n}$. An element $f$ of $Out(F_{n})$ is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by $f$. The existence in $Out(F_{n}), n > 2$, of elements with non-linear polynomial growth is a feature of $Out(F_{n})$ not shared by mapping class groups of surfaces.
To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f $ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.
Keywords : Out(Fn); conjugacy problem; train tracks
MSC Codes : To avoid some finite order behavior, we restrict attention to the subset $UPG(F_{n})$ of $Out(F_{n})$ consisting of polynomially growing elements whose action on $H_{1}(F_{n}, Z)$ is unipotent. In particular, if $f$ is polynomially growing and acts trivially on $H_{1}(F_{n}, Z_{3})$ then $f $ is in $UPG(F_{n})$ and further every polynomially growing element of $Out(F_{n})$ has a power that is in $UPG(F_{n})$. The goal of the talk is to describe an algorithm to decide given $f,g$ in $UPG(F_{n})$ whether or not there is h in $Out(F_{n})$ such that $hf h^{-1} = g$.
The conjugacy problem for linearly growing elements of $UPG(F_{n})$ was solved by Cohen-Lustig. Krstic-Lustig-Vogtmann solved the case of linearly growing elements of $Out(F_{n})$.
A key technique is our use of train track representatives for elements of $Out(F_{n})$, a method pioneered by Bestvina-Handel in the early 1990s that has since been ubiquitous in the study of $Out(F_{n})$.
Keywords : Out(Fn); conjugacy problem; train tracks
20F65 - Geometric group theory
57M07 - Topological methods in group theory
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 18/07/2019 Conference Date : 17/06/2019 Subseries : Research talks arXiv category : Group Theory Mathematical Area(s) : Algebra ; Topology Format : MP4 (.mp4) - HD Video Time : 00:55:29 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-06-17_Feighn.mp4 
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Information on the Event
Event Title : Aspects of Non-Positive and Negative Curvature in Group Theory / Courbure négative et courbure négative ou nulle en théorie des groupesEvent Organizers : Bromberg, Kenneth ; Hilion, Arnaud ; Kazachkov, Ilya ; Sageev, Michah ; Tao, Jing Dates : 17/06/2019 - 21/06/2019 Event Year : 2019 Event URL : https://conferences.cirm-math.fr/1958.html
Citation Data
DOI : 10.24350/CIRM.V.19539203Cite this video as: Feighn, Mark (2019). The conjugacy problem for polynomially growing elements of $Out(F_{n})$. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19539203 URI : http://dx.doi.org/10.24350/CIRM.V.19539203 |
See Also
- [Multi angle] Counting curves of given type, revisited / Author of the conference Souto, Juan.
- [Multi angle] Torsion groups do not act on 2-dimensional CAT(0) complexes / Author of the conference Przytycki, Piotr.
- [Multi angle] Artin groups and mapping class groups / Author of the conference Hamenstädt, Ursula.
- [Multi angle] Conformal dimension and free by cyclic groups / Author of the conference Algom-Kfir, Yael.
Bibliography
- Feighn, M., & Handel, M. (2019). The conjugacy problem for UPG elements of $ Out (F_n) $. arXiv preprint arXiv:1906.04147. - https://arxiv.org/abs/1906.04147
