An algorithm to detect full irreducibility by bounding the volume of periodic free factors

Clay, Matt; Mangahas, Johanna; Pettet, Alexandra

doi:10.1307/mmj/1434731924

Cited by 5 publications

(5 citation statements)

References 21 publications

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“…In the special case that F 1 = ∅, Corollary 1.2 is an algorithm for checking if ψ is fully irreducible. Our algorithm in this special case is different from the ones given in [Kap14] and [CMP15]. More recently, Kapovich [Kap] has produced a polynomial time algorithm to detect full irreducibility.…”

Section: Introductionmentioning

confidence: 96%

Algorithmic constructions of relative train track maps and CTs

Feighn

Handel

2018

Groups Geom. Dyn.

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Building on [BH92, BFH00], we proved in [FH11] that every element ψ of the outer automorphism group of a finite rank free group is represented by a particularly useful relative train track map. In the case that ψ is rotationless (every outer automorphism has a rotationless power), we showed that there is a type of relative train track map, called a CT, satisfying additional properties. The main result of this paper is that the constructions of these relative train tracks can be made algorithmic. A key step in our argument is proving that it is algorithmic to check if an inclusion F F of φ-invariant free factor systems is reduced. We also give applications of the main result.

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Section: Introductionmentioning

confidence: 96%

Algorithmic constructions of relative train track maps and CTs

Feighn

Handel

2018

Groups Geom. Dyn.

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“…More generally the Guirardel core has been used to study the orbit of cofinite actions on an R-tree by subgroups of Out(F g ) [2], [4], [5], [6], [10]. This makes it particularly well-suited for the problem in this paper, since the Stallings maps for a given Heegaard splitting form an Out(F g ) × Out(F g ) orbit.…”

Section: Propositionmentioning

confidence: 99%

Heegaard splittings and virtually special square complexes

Sadanand¹

2023

Preprint

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We give a new characterization of irreducibility of Heegaard splittings for most 3-manifolds: a Heegaard splitting is irreducible if and only if it has an augmented Heegaard diagram that is a surface. A Heegaard splitting is a decomposition of a closed orientable 3-manifold into two isomorphic handle bodies that have a shared boundary surface. Usually, a number of curves on the shared boundary surface, called a Heegaard diagram, are used to describe a Heegaard splitting. We define a more complete object, the augmented Heegaard diagram, by building on a method used by Stallings to encode the information of a Heegaard splitting.Augmented Heegaard diagrams have several desirable properties: each 2-cell is a square, they have non-positive combinatorial curvature and they are virtually special. Restricting to manifolds that do not have S 1 ×S 2 as a connect summand, augmented Heegaard diagrams lend themselves well to understanding the decomposition of a 3-manifold via connect sum, leading to the characterization of irreducibility above. Along the way to proving this result, we find combinatorial methods to simplify Stallings's description of Heegaard splittings and find a connection to Guirardel's notion of a core.

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“…After our paper [30], Clay, Mangahas and Pettet [12] produced a different algorithm for deciding whether an element of ϕ ∈ Out(F N ) is fully irreducible. Their algorithm was more efficient than that in [30], but did not include an explicit complexity estimate.…”

Section: Introductionmentioning

confidence: 99%

“…An element ϕ ∈ Out(F N ) is called primitively atoroidal if there do not exist n = 0 and a primitive element g ∈ F N such that ϕ n [g] = [g]. This notion, or rather its negation, first appeared in [12], where non primitively atoroidal elements of Out(F N ) are called cyclically reducible. Primitively atoroidal elements of Out(F N ) are also specifically studied in [23].…”

Section: Introductionmentioning

confidence: 99%

Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell

Kapovich

2016

Preprint

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In [30] we produced an algorithm for deciding whether or not an element ϕ ∈ Out(FN ) is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this paper we refine the algorithm from [30] by eliminating these inefficient features, and also by eliminating any use of mapping class groups algorithms.Our main result is to produce, for any fixed N ≥ 3, an algorithm which, given a topological representative f of an element ϕ of Out(FN ), decides in polynomial time in terms of the "size" of f , whether or not ϕ is fully irreducible.In addition, we provide a train track criterion of being fully irreducible which covers all fully irreducible elements of Out(FN ), including both atoroidal and non-atoroidal ones.We also give an algorithm, alternative to that of Turner, for finding all the indivisible Nielsen paths in an expanding train track map, and estimate the complexity of this algorithm.An appendix by Mark Bell provides a polynomial upper bound, in terms of the size of the topological representative, on the complexity of the Bestvina-Handel algorithm[3] for finding either an irreducible train track representative or a topological reduction.

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An algorithm to detect full irreducibility by bounding the volume of periodic free factors

Abstract: We provide an effective algorithm for determining whether an element φ of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.

Cited by 5 publications

References 21 publications

Algorithmic constructions of relative train track maps and CTs

Algorithmic constructions of relative train track maps and CTs

Heegaard splittings and virtually special square complexes

Detecting fully irreducible automorphisms: a polynomial time algorithm. With an appendix by Mark C. Bell

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