Loxodromic elements for the relative free factor complex

Gupta, Radhika

doi:10.1007/s10711-017-0310-5

Cited by 14 publications

(19 citation statements)

References 32 publications

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“…(2) there exists a neighbourhood V of T such that (V )φ p converges to T + φ , uniformly as p → ∞ Proof. The proof goes as in the classical case ([2, Lemma 3.4] and [27, Proposition 6.1]) and is the same as that of [22,Proposition 3.3]. The iwip property is used there, just in order to ensure the existence of a train track representative of the automorphism with primitive transition matrix.…”

Section: Statement Of North-south Dynamicsmentioning

confidence: 97%

“…The so-called North-South dynamics for iwip automorphisms in the classical Culler-Vogtmann Outer space CV n is a well know fact established in [27], and generalised in [22] to a more general case. (In [22] is where the hypothesis rank(G) ≥ 3 is required). Here we need a North-South dynamics for our case, which is slightly more general.…”

Section: Statement Of North-south Dynamicsmentioning

confidence: 99%

“…The proof of Theorem 5.1.1 below, is essentially exactly the same as the proof of [22,Theorem C], where the author assumes that G is free and that the automorphism is iwip (or fully irreducible). However, her proof applies for general groups, where some missing point can be filled using results of [19].…”

Section: Statement Of North-south Dynamicsmentioning

confidence: 99%

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On the action of relatively irreducible automorphisms on their train tracks

Francaviglia,

Martino,

Syrigos

2021

Preprint

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Let G be a group and let G be a free factor system of G, namely a free splitting ofIn this paper, we study the set of train track points for G-irreducible automorphisms φ with exponential growth (relatively to G). Such set is known to coincide with the minimally displaced set Min(φ) of φ.Our main result is that Min(φ) is co-compact, under the action of the cyclic subgroup generated by φ.Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of Min(φ) is in uniform distance from Min(φ −1 ). We also prove that the action of G on the product of the attracting and the repelling trees for φ, is discrete. Finally, we get some fine insight about the local topology of relative outer space.As an application, we generalise a classical result of Bestvina, Feighn and Handel for the centralisers of irreducible automorphisms of free groups, in the more general context of relatively irreducible automorphisms of a free product. We also deduce that centralisers of elements of Out(F 3 ) are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that Min(φ) is quasi-isometric to a line.

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Section: Statement Of North-south Dynamicsmentioning

confidence: 97%

Section: Statement Of North-south Dynamicsmentioning

confidence: 99%

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On the action of relatively irreducible automorphisms on their train tracks

Francaviglia,

Martino,

Syrigos

2021

Preprint

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“…We observe that both S 0 and T 0 are points in the relative outer space O(F, A), which inherits the subspace topology from PO. Moreover, φ is fully irreducible relative to A, and as such, it acts with north-south dynamics on PO(F, A) [Gup16]. Recall that for each i ∈ N,…”

Section: Proposition 35 ([Hm11]mentioning

confidence: 99%

Loxodromics for the cyclic splitting complex and their centralizers

Gupta

Wigglesworth

2019

Pacific J. Math.

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We show that an outer automorphism acts loxodromically on the cyclic splitting complex if and only if it has a filling lamination and no generic leaf of the lamination is carried by a vertex group of a cyclic splitting. This is the analog for the cyclic splitting complex of Handel-Mosher's theorem on loxodromics for the free splitting complex. We also show that such outer automorphisms have virtually cyclic centralizers. of φ. Recently, Algom-Kfir and Pfaff showed [AP16] that centralizers of fully irreducible outer automorphisms with lone axes are isomorphic to Z. We also mention a result of Kapovich and Lustig [KL11]: automorphisms whose limiting trees are free have virtually cyclic centralizers.The main motivation for examining the centralizers of loxodromic elements of FZ (and FS) is to understand which automorphisms have the potential to be WPD elements for the action of Out(F) on these complexes.Corollary 1.4. Any outer automorphism that is loxodromic for the action of Out(F) on FS but elliptic for the action on FZ is not a WPD element for the action on FS.The result that centralizers of loxodromic elements of FZ are virtually cyclic is a promising sign for the following conjecture:Conjecture 1.5. The action of Out(F) on FZ is a WPD action. That is, every loxodromic element for the action satisfies WPD.

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“…In this paper we develop the machinery of currents relative to a free factor system. This machinery is then used in [Gup16] to classify the outer automorphisms that act with positive translation length on a relative free factor complex.…”

Section: Introductionmentioning

confidence: 99%