The conjugacy problem for UPG elements of Out(Fn)

Feighn, Mark¹ ; Handel, Michael²

¹Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, United States
²Mathematics and Computer Science Department, Herbert H Lehman College (CUNY), Bronx, NY, United States

Geometry & topology, Tome 29 (2025) no. 4, pp. 1693-1817

Résumé

An element ϕ of the outer automorphism group Out(Fn) of the rank n free group Fn is polynomially growing if the word lengths of conjugacy classes in Fn grow at most polynomially under iteration by ϕ. It is unipotent if, additionally, its action on the first homology of Fn 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(Fn) for the subset of unipotent elements. Specifically, there is an algorithm that decides if two such are conjugate in Out(Fn).

Détail

DOI : 10.2140/gt.2025.29.1693

Classification : 20F65, 57M07
Keywords: automorphisms of free groups, conjugacy problem, train tracks, rotationless, unipotent

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