The conjugacy problem for UPG elements of Out(Fn)

Feighn, Mark; Handel, Michael

doi:10.2140/gt.2025.29.1693

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The conjugacy problem for UPG elements of $\mathrm{Out}(F_n)$

Mark Feighn and Michael Handel

Geometry & Topology 29 (2025) 1693–1817

DOI: 10.2140/gt.2025.29.1693

Abstract

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

Keywords

automorphisms of free groups, conjugacy problem, train tracks, rotationless, unipotent

Mathematical Subject Classification 2010

Primary: 20F65, 57M07

References

Publication

Received: 2 July 2019

Revised: 7 March 2022

Accepted: 30 May 2024

Published: 27 June 2025

Proposed: Martin R Bridson

Seconded: David Fisher, Bruce Kleiner

Authors

Mark Feighn
Department of Mathematics and Computer Science Rutgers University Newark, NJ United States

Michael Handel
Mathematics and Computer Science Department Herbert H Lehman College (CUNY) Bronx, NY United States

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Volume 29, issue 4 (2025)