Polynomial growth and subgroups of $\mathrm{Out}\left(F_N\right)$

Guerch, Yassine

Annales Henri Lebesgue, Tome 6 (2023), pp. 595-625

This paper, which is the last of a series of three papers, studies dynamical properties of elements of $\mathrm{Out}\left(F_N\right)$, the outer automorphism group of a nonabelian free group $F_N$. We prove that, for every subgroup $H$ of $\mathrm{Out}\left(F_N\right)$, there exists an element $\varphi \in H$ such that, for every element $g$ of $F_N$, the conjugacy class $\left[g\right]$ has polynomial growth under iteration of $\varphi$ if and only if $\left[g\right]$ has polynomial growth under iteration of every element of $H$.

Dans cet article, nous étudions des propriétés dynamiques des éléments de $\mathrm{Out}\left(F_N\right)$, le groupe des automorphismes extérieurs d’un groupe non abélien libre $F_N$ de rang $N\ge 2$. Nous montrons que, pour tout sous-groupe $H$ de $\mathrm{Out}\left(F_N\right)$, il existe un élément $\varphi \in H$, appelé dynamiquement générique, qui capture la croissance polynomiale de $H$ au sens suivant. La classe de conjugaison d’un élément $g\in F_N$ est à croissance polynomiale sous itération de tous les éléments de $H$ si, et seulement si, la classe de conjugaison de $g$ est à croissance polynomiale sous itération de $\varphi$.

Classification : 20E05, 20E08, 20E36, 20F65
Keywords: Nonabelian free groups, outer automorphism groups, space of currents, group actions on trees

[BFH05] Bestvina, Mladen; Feighn, Mark; Handel, Michael The Tits alternative for Out$\left({\mathrm{F}}_n\right)$ II: A Kolchin type theorem, Ann. Math., Volume 161 (2005) no. 1, pp. 1-59 | DOI | Zbl | MR

[BH92] Bestvina, Mladen; Handel, Michael Train tracks and automorphisms of free groups (2), Volume 135, 1992 no. 1, pp. 1-51 | DOI | Zbl | MR

[CU18] Clay, Matthew; Uyanik, Caglar Simultaneous construction of hyperbolic isometries, Volume 294, 2018 no. 1, pp. 71-88 | DOI | Zbl | MR

[CU20] Clay, Matthew; Uyanik, Caglar Atoroidal dynamics of subgroups of $\mathrm{Out}\left(F_N\right)$ (2), Volume 102, 2020 no. 2, pp. 818-845 | DOI | Zbl | MR

[FH11] Feighn, Mark; Handel, Michael The recognition theorem for $Out\left(F_n\right)$, Groups Geom. Dyn., Volume 5 (2011) no. 1, pp. 39-106 | DOI | Zbl | MR

[FLP79] Travaux de Thurston sur les surfaces (Fathi, Albert; Laudenbach, François; Poenaru, Valentin, eds.), Astérisque, 66-67, Société Mathématique de France, 1979 | Zbl

[FM11] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2011 | Zbl | DOI

[Fuj02] Fujiwara, Koji On the outer automorphism group of a hyperbolic group, Isr. J. Math., Volume 131 (2002), pp. 277-284 | Zbl | MR | DOI

[GaL95] Gaboriau, Damien; Levitt, Gilbert The rank of actions on $ℝ$-trees, Ann. Sci. Éc. Norm. Supér., Volume 28 (1995) no. 5, pp. 549-570 | MR | Zbl | DOI

[GH19] Guirardel, Vincent; Horbez, Camille Algebraic laminations for free products and arational trees, Algebr. Geom. Topol., Volume 19 (2019) no. 5, pp. 2283-2400 | MR | Zbl | DOI

[GH22] Guirardel, Vincent; Horbez, Camille Boundaries of relative factor graphs and subgroup classification for automorphisms of free products, Geom. Topol., Volume 26 (2022) no. 1, pp. 71-126 | Zbl | MR | DOI

[GL15] Guirardel, Vincent; Levitt, Gilbert McCool groups of toral relatively hyperbolic groups, Algebr. Geom. Topol., Volume 15 (2015) no. 6, pp. 3485-3534 | MR | Zbl | DOI

[GL17] Guirardel, Vincent; Levitt, Gilbert JSJ decompositions of groups, Astérisque, 395, Société Mathématique de France, 2017 | Zbl

[Gue21] Guerch, Yassine Currents relative to a malnormal subgroup system (2021) (https://arxiv.org/abs/2112.01112)

[Gue22a] Guerch, Yassine Géométrie, dynamique et rigidité de groupes d’automorphismes de produits libres, Ph. D. Thesis, Université Paris-Saclay, Paris, France (2022)

[Gue22b] Guerch, Yassine North-South type dynamics of relative atoroidal automorphisms of free groups (2022) (https://arxiv.org/abs/2203.04112)

[Gup17] Gupta, Radhika Relative currents, Conform. Geom. Dyn., Volume 21 (2017), pp. 319-352 | MR | Zbl | DOI

[Gup18] Gupta, Radhika Loxodromic elements for the relative free factor complex, Geom. Dedicata, Volume 196 (2018), pp. 91-121 | MR | Zbl | DOI

[HM20] Handel, Michael; Mosher, Lee Subgroup Decomposition in $\mathrm{Out}\left(F_n\right)$, Memoirs of the American Mathematical Society, 1280, American Mathematical Society, 2020 | Zbl | DOI

[Hor16] Horbez, Camille Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings, J. Topol., Volume 9 (2016) no. 2, pp. 401-450 | MR | Zbl | DOI

[Iva92] Ivanov, Nikolai V. Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, 115, American Mathematical Society, 1992 | MR | Zbl

[Lev09] Levitt, Gilbert Counting growth types of automorphisms of free groups, Geom. Funct. Anal., Volume 19 (2009) no. 4, pp. 1119-1146 | MR | Zbl | DOI

[Man14a] Mann, Brian Hyperbolicity of the cyclic splitting graph, Geom. Dedicata, Volume 173 (2014), pp. 271-280 | MR | Zbl | DOI

[Man14b] Mann, Brian Some hyperbolic $\mathrm{Out}\left(F_N\right)$-graphs and nonunique ergodicity of very small $F_N$-trees, Ph. D. Thesis, University of Utah, Salt Lake City, USA (2014)

[Mar95] Martin, Reiner Non-uniquely ergodic foliations of thin-type, measured currents and automorphisms of free groups, Ph. D. Thesis, University of California, Los Angeles, USA (1995) | MR

[McC85] McCarthy, John A “Tits-alternative” for subgroups of surface mapping class groups, Trans. Am. Math. Soc., Volume 291 (1985), pp. 583-612 | MR | Zbl | DOI

[SW79] Scott, G. Peter; Wall, Charles T. C. Topological methods in group theory, Homological group theory, Proc. Symp., Durham 1977 (Wall, C. T. C., ed.) (London Mathematical Society Lecture Note Series), Volume 36, Cambridge University Press, 1979, pp. 137-203 | MR | Zbl

[Tit72] Tits, Jacques Free subgroups in linear goups, J. Algebra, Volume 20 (1972), pp. 250-270 | Zbl | DOI

[ZVC80] Zieschang, Heiner; Vogt, Elmar; Coldewey, Hans-Dieter Surfaces and planar discontinuous groups, Lecture Notes in Mathematics, 835, Springer, 1980 | Zbl | DOI