##
**Hyperbolic extensions of free groups from atoroidal ping-pong.**
*(English)*
Zbl 1444.20023

By pioneer works of M. Gromov in geometric group theory, the notion of hyperbolic group was introduced in the 1980s, and his works made a great influence on the development of group theory. His paper [Publ. Math. Sci. Res. Inst. 8, 75–263 (1987; Zbl 0634.20015)] is referred by a huge number of authors. So far, hyperbolic groups have been studied from various perspectives, for example characterization by actions, presentations, classifying spaces, decision problems, growth functions, boundaries and so on. For example, see Section 7 of a remarkable survey article [Münster J. Math. 1, No. 1, 73–108 (2008; Zbl 1197.20036)] by W. Lück, who describes himself as “a non-expert who likes geometric group theory” in the article.

The study of hyperbolic extensions of surface groups by mapping class groups goes back to Thurston’s work on the hyperbolization of fibered 3-manifolds. It has achieved a good progress by several authors, including M. Bestvina and M. Feighn [J. Differ. Geom. 35, No. 1, 85–102 (1992; Zbl 0724.57029)], B. Farb and L. Mosher [Geom. Topol. 6, 91–152 (2002; Zbl 1021.20034)] and U. Hamenstädt [“Word hyperbolic extensions of surface groups”, Preprint, arXiv:math.GT/0505244]. As a comparative study, hyperbolic extensions of free groups by outer automorphism groups of free groups has been also studied by S. Dowdall and S. Taylor [Geom. Topol. 22, No. 1, 517–570 (2018; Zbl 1439.20034)]. Let \(F_n\) be a free group of rank \(n \geq 3\). The quotient group of the automorphism group \(\operatorname{Aut} F_n\) by the inner automorphism group \(\mathrm{Inn}\,F_n\) is the outer automorphism group of \(F_n\), denoted by \(\operatorname{Out}F_n\). Let \(\pi : \operatorname{Aut} F_n \rightarrow \operatorname{Out}F_n\) be the natural map. Then we have a question:

What is the necessary and sufficient condition for a subgroup \(\Gamma \leq \operatorname{Out}F_n\) to satisfy that the preimage \(\pi^{-1}(\Gamma)\) is hyperbolic?

Dowdall and Taylor [loc. cit.] initiated a systematic study to attack this problem. In the present paper, the author develops their research and gives new examples of hyperbolic extensions. Let us explain it more precisely. An outer automorphism \(\varphi \in \operatorname{Out}F_n\) is called atroidal (fully irreducible) if no power of \(\varphi\) fixes a nontrivial conjugacy class (the conjugacy class of a proper free factor) of \(F_n\). A subgroup of \(\operatorname{Out}F_n\) is called purely atroidal if every infinite-order element is atroidal. In Corollary 1.5, the author shows that for a fully irreducible atroidal outer automorphism \(\varphi \in \operatorname{Out}F_n\) and for any atroidal outer automorphism \(\psi \in \operatorname{Out}F_n\) which is not commensurable with \(\varphi\) there exists an \(M>0\) such that for all \(n, m >M\) the subgroup \(\Gamma \leq \operatorname{Out}F_n\) generated by \(\varphi^n\) and \(\psi^m\) is purely atroidal, and \(\pi^{-1}(\Gamma)\) is hyperbolic.

In this paper, in Section 3 the author proves that all atroidal outer automorphisms of \(F_n\) act on the space of projectivized geodesic currents with generalized north-south dynamics (see Theorem 1.4.). This part seems to be the main part of the paper. Using Theorem 1.4, the author shows a proposition with respect to a certain property of atroidal outer automorphisms of \(F_n\) (see Proposition 4.4.). Then using Proposition 4.4 and the Bestvina-Feighn combination theorem, the author proves the main theorem of the paper, which is a generalization of a result of M. Bestvina et al. [Geom. Funct. Anal. 7, No. 2, 215–244 (1997; Zbl 0884.57002)] (see Theorem 1.1.). As a corollary to Theorem 1.1, the author proves Corollary 1.5.

The study of hyperbolic extensions of surface groups by mapping class groups goes back to Thurston’s work on the hyperbolization of fibered 3-manifolds. It has achieved a good progress by several authors, including M. Bestvina and M. Feighn [J. Differ. Geom. 35, No. 1, 85–102 (1992; Zbl 0724.57029)], B. Farb and L. Mosher [Geom. Topol. 6, 91–152 (2002; Zbl 1021.20034)] and U. Hamenstädt [“Word hyperbolic extensions of surface groups”, Preprint, arXiv:math.GT/0505244]. As a comparative study, hyperbolic extensions of free groups by outer automorphism groups of free groups has been also studied by S. Dowdall and S. Taylor [Geom. Topol. 22, No. 1, 517–570 (2018; Zbl 1439.20034)]. Let \(F_n\) be a free group of rank \(n \geq 3\). The quotient group of the automorphism group \(\operatorname{Aut} F_n\) by the inner automorphism group \(\mathrm{Inn}\,F_n\) is the outer automorphism group of \(F_n\), denoted by \(\operatorname{Out}F_n\). Let \(\pi : \operatorname{Aut} F_n \rightarrow \operatorname{Out}F_n\) be the natural map. Then we have a question:

What is the necessary and sufficient condition for a subgroup \(\Gamma \leq \operatorname{Out}F_n\) to satisfy that the preimage \(\pi^{-1}(\Gamma)\) is hyperbolic?

Dowdall and Taylor [loc. cit.] initiated a systematic study to attack this problem. In the present paper, the author develops their research and gives new examples of hyperbolic extensions. Let us explain it more precisely. An outer automorphism \(\varphi \in \operatorname{Out}F_n\) is called atroidal (fully irreducible) if no power of \(\varphi\) fixes a nontrivial conjugacy class (the conjugacy class of a proper free factor) of \(F_n\). A subgroup of \(\operatorname{Out}F_n\) is called purely atroidal if every infinite-order element is atroidal. In Corollary 1.5, the author shows that for a fully irreducible atroidal outer automorphism \(\varphi \in \operatorname{Out}F_n\) and for any atroidal outer automorphism \(\psi \in \operatorname{Out}F_n\) which is not commensurable with \(\varphi\) there exists an \(M>0\) such that for all \(n, m >M\) the subgroup \(\Gamma \leq \operatorname{Out}F_n\) generated by \(\varphi^n\) and \(\psi^m\) is purely atroidal, and \(\pi^{-1}(\Gamma)\) is hyperbolic.

In this paper, in Section 3 the author proves that all atroidal outer automorphisms of \(F_n\) act on the space of projectivized geodesic currents with generalized north-south dynamics (see Theorem 1.4.). This part seems to be the main part of the paper. Using Theorem 1.4, the author shows a proposition with respect to a certain property of atroidal outer automorphisms of \(F_n\) (see Proposition 4.4.). Then using Proposition 4.4 and the Bestvina-Feighn combination theorem, the author proves the main theorem of the paper, which is a generalization of a result of M. Bestvina et al. [Geom. Funct. Anal. 7, No. 2, 215–244 (1997; Zbl 0884.57002)] (see Theorem 1.1.). As a corollary to Theorem 1.1, the author proves Corollary 1.5.

Reviewer: Takao Satoh (Tokyo)

### MSC:

20F28 | Automorphism groups of groups |

20F67 | Hyperbolic groups and nonpositively curved groups |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

20E05 | Free nonabelian groups |

57M07 | Topological methods in group theory |

### Citations:

Zbl 0634.20015; Zbl 1197.20036; Zbl 0724.57029; Zbl 1021.20034; Zbl 0884.57002; Zbl 1439.20034
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\textit{C. Uyanik}, Algebr. Geom. Topol. 19, No. 3, 1385--1411 (2019; Zbl 1444.20023)

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