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H 2 The conjugacy problem for polynomially growing elements of $Out(F_{n})$

Auteurs : Feighn, Mark (Auteur de la Conférence)
CIRM (Editeur )

 Loading the player... Conjugacy problem for Out (Fn) Eigengraphs and invariants Whitehead’s theorem and reduction Algebraic invariants Induction Understanding rays

Résumé : (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

Codes MSC :
20F65 - Geometric group theory
57M07 - Topological methods in group theory

 Informations sur la Vidéo Réalisateur : Hennenfent, Guillaume Langue : Anglais Date de publication : 18/07/2019 Date de captation : 17/06/2019 Collection : Research talks ; Algebra ; Topology Format : MP4 (.mp4) - HD Durée : 00:55:29 Domaine : Algebra ; Topology Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2019-06-17_Feighn.mp4 Informations sur la rencontre Nom de la rencontre : Aspects of Non-Positive and Negative Curvature in Group Theory / Courbure négative et courbure négative ou nulle en théorie des groupesOrganisateurs de la rencontre : Bromberg, Kenneth ; Hilion, Arnaud ; Kazachkov, Ilya ; Sageev, Michah ; Tao, JingDates : 17/06/2019 - 21/06/2019 Année de la rencontre : 2019 URL Congrès : https://conferences.cirm-math.fr/1958.html Citation Data DOI : 10.24350/CIRM.V.19539203 Cite 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

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