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The conjugacy problem for Out(F3)

The conjugacy problem for Out(F3)
The conjugacy problem for Out(F3)
We present a solution to the Conjugacy Problem in the group of outer-automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth, and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer-automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer-automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.
math.GR
Dahmani, François
4310b34a-9333-4887-8767-bd76ef9152ad
Francaviglia, Stefano
91be45eb-fadf-48ed-abe8-107c65f85c6c
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Touikan, Nicholas
e948553e-a846-4fbb-8291-a2e5f93c82dd
Dahmani, François
4310b34a-9333-4887-8767-bd76ef9152ad
Francaviglia, Stefano
91be45eb-fadf-48ed-abe8-107c65f85c6c
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Touikan, Nicholas
e948553e-a846-4fbb-8291-a2e5f93c82dd

[Unknown type: UNSPECIFIED]

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Abstract

We present a solution to the Conjugacy Problem in the group of outer-automorphisms of $F_3$, a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth, and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer-automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer-automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

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2311.04010v1 - Author's Original
Available under License Creative Commons Public Domain Dedication.
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Published date: 7 November 2023
Additional Information: 27 pages
Keywords: math.GR

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Local EPrints ID: 485035
URI: http://eprints.soton.ac.uk/id/eprint/485035
DOI: doi:10.48550/arXiv.2311.04010
PURE UUID: 4c5a0325-b28e-4d98-8b99-86f05874d54f
ORCID for Armando Martino: ORCID iD orcid.org/0000-0002-5350-3029

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Date deposited: 28 Nov 2023 17:50
Last modified: 18 Mar 2024 03:11

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Author: François Dahmani
Author: Stefano Francaviglia
Author: Armando Martino ORCID iD
Author: Nicholas Touikan

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