[Sur l’action des automorphismes irréductibles sur leurs train tracks]
Let $G$ be a group and let $\mathcal{G}$ be a free factor system of $G$, namely a free splitting of $G$ as $G=G_1* \cdots *G_k*F_r$. In this paper, we study the set of train track points for $\mathcal{G}$-irreducible automorphisms $\phi $ with exponential growth. Such set is known to coincide with the minimally displaced set $\operatorname{Min}(\phi )$ of $\phi $, in the relative deformation space corresponding to the splitting. The theory of such relative spaces, even if it is more general by its own nature, is crucial to understanding reducible automorphisms of free groups, as any automorphism is relatively irreducible with respect to some free factor system $\mathcal{G}$.
Our main result is that $\operatorname{Min}(\phi )$ is co-compact, under the action of the cyclic subgroup generated by $\phi $.
Along the way we obtain other results that could be of independent interest. For instance, we prove that any point of $\operatorname{Min}(\phi )$ is in uniform distance from $\operatorname{Min}(\phi ^{-1})$. We also prove that the action of $G$ on the product of the attracting and the repelling trees for $\phi $, is discrete. Finally, we get some fine insight about the local topology of relative outer space.
Some applications of co-compactness are discussed. In particular 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. From this, we deduce that centralisers of elements of $\operatorname{Out}(F_3)$ are finitely generated, which was previously unknown. Finally, we mention that an immediate corollary of co-compactness is that the set $\operatorname{Min}(\phi )$ is always quasi-isometric to a line.
Soit $G$ un groupe et $\mathcal{G}$ un système de facteurs libres de $G$, c’est a dire une décomposition de $G$ en produits libres comme $G = G_1 * \cdots * G_k * F_r$. Dans cet article, nous étudions l’ensemble de points train track pour les automorphismes $\phi $ qui sont $\mathcal{G}$-irréductibles et à croissance exponentielle. On sait qu’un tel ensemble coïncide avec l’ensemble $\operatorname{Min}(\phi )$ des points qui sont minimalement déplacé par $\phi $ dans l’espace de déformation relatif correspondant à la décomposition $\mathcal{G}$. La théorie de tels espaces relatifs, même si elle est plus générale par nature, est essentielle pour comprendre les automorphismes réductibles des groupes libres, car tout automorphisme est relativement irréductible par rapport à quelque système de facteurs libres $\mathcal{G}$.
Notre résultat principal est que l’action sur $\operatorname{Min}(\phi )$ du groupe cyclique engendré par $\phi $ est co-compacte.
En passant, nous obtenons d’autres résultats qui pourraient présenter un intérêt indépendant. Par exemple, nous prouvons que tout point de $\operatorname{Min}(\phi )$ est à distance uniforme de $\operatorname{Min}(\phi ^{-1})$. Nous prouvons également que l’action de $G$ sur le produit des arbres attractif et répulsif de $\phi $ est discrète. Enfin, nous obtenons un aperçu précis de la topologie locale de l’outre-espace relatif.
Quelques applications de la co-compacité sont discutées. En particulier, nous généralisons un résultat classique de Bestvina, Feighn et Haendel pour les centralisateurs des automorphismes irréductibles du groupe libre, dans le contexte plus général des automorphismes relativement irréductibles de produits libres. On en déduit que les centralisateurs des éléments de $\operatorname{Out}(F_3)$ sont finiment engendrés, ce qui était auparavant inconnu. Finalement, nous mentionnons qu’un corollaire immédiat de la co-compacité est que l’ensemble $\operatorname{Min}(\phi )$ est toujours quasi-isométrique à une droite.
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Keywords: automorphisms of free groups, conjugacy problem
Mots-clés : automorphismes des groupes libres, problème de la conjugaison
Francaviglia, Stefano 1 ; Martino, Armando 2 ; Syrigos, Dionysios 2
@unpublished{AIF_0__0_0_A35_0,
author = {Francaviglia, Stefano and Martino, Armando and Syrigos, Dionysios},
title = {On the action of relatively irreducible automorphisms on their train tracks},
journal = {Annales de l'Institut Fourier},
year = {2026},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
doi = {10.5802/aif.3747},
language = {en},
note = {Online first},
}
TY - UNPB AU - Francaviglia, Stefano AU - Martino, Armando AU - Syrigos, Dionysios TI - On the action of relatively irreducible automorphisms on their train tracks JO - Annales de l'Institut Fourier PY - 2026 PB - Association des Annales de l’institut Fourier N1 - Online first DO - 10.5802/aif.3747 LA - en ID - AIF_0__0_0_A35_0 ER -
%0 Unpublished Work %A Francaviglia, Stefano %A Martino, Armando %A Syrigos, Dionysios %T On the action of relatively irreducible automorphisms on their train tracks %J Annales de l'Institut Fourier %D 2026 %V 0 %N 0 %I Association des Annales de l’institut Fourier %Z Online first %R 10.5802/aif.3747 %G en %F AIF_0__0_0_A35_0
Francaviglia, Stefano; Martino, Armando; Syrigos, Dionysios. On the action of relatively irreducible automorphisms on their train tracks. Annales de l'Institut Fourier, Online first, 92 p.
[1] Centralisers of linear growth automorphisms of free groups (2022) | arXiv
[2] Laminations, trees, and irreducible automorphisms of free groups, Geom. Funct. Anal., Volume 7 (1997) no. 2, pp. 215-244 | Zbl | DOI | MR
[3] The Tits alternative for I: Dynamics of exponentially-growing automorphisms, Ann. Math., Volume 151 (2000) no. 2, pp. 517-623 | DOI | Zbl
[4] Train tracks and automorphisms of free groups, Ann. Math., Volume 135 (1992) no. 1, pp. 1-51 | DOI | Zbl
[5] The conjugacy problem for Dehn twist automorphisms of free groups, Comment. Math. Helv., Volume 74 (1999) no. 2, pp. 179-200 | Zbl | MR | DOI
[6] Group actions on -trees, Proc. Lond. Math. Soc., Volume 55 (1987), pp. 571-604 | Zbl | DOI | MR
[7] Relative hyperbolicity for automorphisms of free products and free groups, J. Topol. Anal., Volume 14 (2022) no. 1, pp. 55-92 | DOI | MR | Zbl
[8] Dynamics on free-by-cyclic groups, Geom. Topol., Volume 19 (2015) no. 5, pp. 2801-2899 | Zbl | DOI | MR
[9] Metric properties of outer space, Publ. Mat., Barc., Volume 55 (2011) no. 2, pp. 433-473 | DOI | Zbl | MR
[10] The isometry group of outer space, Adv. Math., Volume 231 (2012) no. 3-4, pp. 1940-1973 | DOI | Zbl | MR
[11] Stretching factors, metrics and train tracks for free products, Ill. J. Math., Volume 59 (2015) no. 4, pp. 859-899 | Zbl | MR
[12] Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks, Trans. Am. Math. Soc., Volume 374 (2021), pp. 3215-3264 | DOI | Zbl | MR
[13] Displacements of automorphisms of free groups II: Connectedness of level sets, Trans. Am. Math. Soc., Volume 375 (2022), pp. 2511-2551 | Zbl | MR
[14] The minimally displaced set of an irreducible automorphism is locally finite, Glas. Mat., III. Ser., Volume 55 (2020) no. 2, pp. 301-336 | DOI | Zbl | MR
[15] The minimally displaced set of an irreducible automorphism of is co-compact, Arch. Math., Volume 116 (2021) no. 4, pp. 369-383 | DOI | MR | Zbl
[16] Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (Niblo, Graham A.; Roller, Martin A., eds.) (London Mathematical Society Lecture Note Series), Volume 182, Cambridge University Press, 1993, pp. 1-295 | MR | Zbl
[17] Approximations of stable actions on -trees, Comment. Math. Helv., Volume 73 (1998), pp. 89-121 | DOI | MR | Zbl
[18] Algebraic laminations for free products and arational trees, Algebr. Geom. Topol., Volume 19 (2019) no. 5, pp. 2283-2400 | DOI | Zbl | MR
[19] Measure equivalence rigidity of (2021) | arXiv | Zbl
[20] The outer space of a free product, Proc. Lond. Math. Soc. (3), Volume 94 (2007) no. 3, pp. 695-714 | DOI | MR | Zbl
[21] Loxodromic Elements in the relative free factor complex, Geom. Dedicata, Volume 196 (2018), pp. 91-121 | DOI | Zbl | MR
[22] Axes in outer space, Memoirs of the American Mathematical Society, 213, American Mathematical Society, 2011 no. 1004, vi+104 pages | DOI | MR | Zbl
[23] Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings, J. Topol., Volume 9 (2016), pp. 401-450 | DOI | Zbl | MR
[24] The boundary of the outer space of a free product, Isr. J. Math., Volume 221 (2017), pp. 179-234 | DOI | Zbl | MR
[25] Automorphism group of a free group: centralizers and stabilizers, J. Algebra, Volume 150 (1992) no. 2, pp. 453-502 | DOI | Zbl | MR
[26] An equivariant Whitehead algorithm and conjugacy for roots of Dehn twist automorphisms, Proc. Edinb. Math. Soc., II. Ser., Volume 44 (2001) no. 1, pp. 117-141 | DOI | MR | Zbl
[27] Irreducible automorphisms of have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu, Volume 2 (2003) no. 1, pp. 59-72 | DOI | MR | Zbl
[28] Irreducible nonsurjective endomorphisms of are hyperbolic, Bull. Lond. Math. Soc., Volume 52 (2020) no. 5, pp. 960-976 | DOI | MR | Zbl
[29] Axioms for translation length functions, Arboreal group theory (Berkeley, CA, 1988) (Mathematical Sciences Research Institute Publications), Volume 19, Springer, 1991, pp. 295-330 | DOI | MR | Zbl
[30] The Gromov topology on -trees, Topology Appl., Volume 32 (1989) no. 3, pp. 197-221 | DOI | MR | Zbl
[31] Trees, Springer, 1980, ix+142 pages | MR | Zbl | DOI
[32] Irreducible laminations for IWIP automorphisms of free products and centralisers (2014) | arXiv | Zbl
[33] Asymmetry of outer space of a free product, Commun. Algebra, Volume 46 (2018) no. 8, pp. 3442-3460 | DOI | MR | Zbl
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