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Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

François Dahmani, Nicholas Touikan

TL;DR

This work reduces the conjugacy problem in ${\mathrm{Out}(G)}$ (notably ${\mathrm{Out}}(F_n)$) to algorithmic questions about mapping tori of polynomially growing automorphisms within a relatively hyperbolic framework. By constructing maximal polynomial-growth sub-mapping-tori and exploiting JSJ decompositions, Dehn fillings (including fibered fillings) and the small modular group, the authors develop a robust strategy to certify isomorphisms and distinguish conjugacy classes through fiber-and-orientation preserving data. The main theorem shows that, under a suite of computability properties for the relevant subgroups and their edge groups, the fiber-and-orientation preserving isomorphism problem for mapping tori is solvable, hence the conjugacy problem in ${\mathrm{Out}(G)}$ is solvable for the considered class of automorphisms. The paper applies the framework to several classes, including atoroidal, almost toral, and unipotent polynomial-growth automorphisms of free groups, providing new decidability results and outlining a path toward a full resolution of the conjugacy problem in ${\mathrm{Out}}(F_n)$. This advances the interface between geometric group theory and algorithmic problems by linking JSJ-type decompositions, Dehn fillings, and Whitehead-type problems to concrete decision procedures in outer automorphism groups.

Abstract

We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path toward a full solution to the conjugacy problem for $Out (F_n)$.

Reducing the conjugacy problem for relatively hyperbolic automorphisms to peripheral components

TL;DR

This work reduces the conjugacy problem in (notably ) to algorithmic questions about mapping tori of polynomially growing automorphisms within a relatively hyperbolic framework. By constructing maximal polynomial-growth sub-mapping-tori and exploiting JSJ decompositions, Dehn fillings (including fibered fillings) and the small modular group, the authors develop a robust strategy to certify isomorphisms and distinguish conjugacy classes through fiber-and-orientation preserving data. The main theorem shows that, under a suite of computability properties for the relevant subgroups and their edge groups, the fiber-and-orientation preserving isomorphism problem for mapping tori is solvable, hence the conjugacy problem in is solvable for the considered class of automorphisms. The paper applies the framework to several classes, including atoroidal, almost toral, and unipotent polynomial-growth automorphisms of free groups, providing new decidability results and outlining a path toward a full resolution of the conjugacy problem in . This advances the interface between geometric group theory and algorithmic problems by linking JSJ-type decompositions, Dehn fillings, and Whitehead-type problems to concrete decision procedures in outer automorphism groups.

Abstract

We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path toward a full solution to the conjugacy problem for .

Paper Structure

This paper contains 17 sections, 23 theorems, 10 equations, 1 figure.

Table of Contents

  1. Preliminaries
  2. Graphs, graphs of groups, and their fundamental groups
  3. Automorphisms of graphs of groups
  4. The main result
  5. General statement
  6. JSJ decompositions
  7. On white vertices: Congruences and Dehn fillings for lists
  8. Peripheral structures, orbits of markings
  9. White vertices of the JSJ decompositions and peripheral structures
  10. Dehn fillings
  11. Back to lists for white vertices
  12. On black vertices: Navigating the Mixed Whitehead Problem
  13. Parts of the fibre in vertex groups
  14. The rest of the fibre and completing the proof of Theorem \ref{['thm;IP_fop']}.
  15. Trivial outer automorphisms of free groups
  16. ...and 2 more sections

Key Result

Theorem 1

Let $G$ be a residually finite torsion-free hyperbolic group, and consider a class $\calO\calA$ of outer automorphisms of $G$. Let $\calP$ be the collection of all maximal polynomial growth sub-mapping-tori given by the elements in $\calO\calA$ and let $\calP'$ be the class of their subgroups that a Then the conjugacy problem in ${\mathrm{Out}\left(G\right)}$ for elements in class $\calO\calA$ is

Figures (1)

  • Figure 1: An automorphism carried by a homotopy equivalence that restricts to pseudo-Anosov maps on surfaces with or without boundary. A remaining free factor is not preserved and all its elements grow exponentially.

Theorems & Definitions (46)

  • Theorem 1
  • Proposition 2
  • Example 3
  • Example 4
  • Corollary 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Corollary 10
  • ...and 36 more