Boundaries of relative factor graphs and subgroup classification for automorphisms of free products

Guirardel, Vincent¹ ; Horbez, Camille²

¹ Institut de recherche en mathématiques de Rennes, UMR 6625, Université de Rennes 1 et CNRS, Rennes, France
² Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, Orsay, France

Geometry & topology, Tome 26 (2022) no. 1, pp. 71-126.

Résumé

Given a countable group $G$ splitting as a free product $G=G_1\ast \cdots \ast G_k\ast F_N$, we establish classification results for subgroups of the group $\mathrm{Out}(G,\mathcal{ℱ})$ of all outer automorphisms of $G$ that preserve the conjugacy class of each $G_i$. We show that every finitely generated subgroup $H\subseteq \mathrm{Out}(G,\mathcal{ℱ})$ either contains a relatively fully irreducible automorphism, or else it virtually preserves the conjugacy class of a proper free factor relative to the decomposition (the finite generation hypothesis on $H$ can be dropped for $G=F_N$, or more generally when $G$ is toral relatively hyperbolic). In the first case, either $H$ virtually preserves a nonperipheral conjugacy class in $G$, or else $H$ contains an atoroidal automorphism. The key geometric tool to obtain these classification results is a description of the Gromov boundaries of relative versions of the free factor graph $\mathrm{FF}$ and the $\mathcal{𝒵}$–factor graph $\mathcal{𝒵}F$, as spaces of equivalence classes of arational trees and relatively free arational trees, respectively. We also identify the loxodromic isometries of $\mathrm{FF}$ with the fully irreducible elements of $\mathrm{Out}(G,\mathcal{ℱ})$, and loxodromic isometries of $\mathcal{𝒵}F$ with the fully irreducible atoroidal outer automorphisms.

Détail

Classification : 20E06, 20E07, 20E08, 20E36
Keywords: automorphism groups of free groups and free products, subgroup classification, Gromov hyperbolic spaces, Gromov boundaries, free factor graph

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