The recognition theorem for \(\text{Out}(F_n)\).

*(English)*Zbl 1239.20036Let \(F_n\) be the free group of rank \(n\). For an automorphism \(\Phi\) of \(F_n\) let \(\varphi\) denote the corresponding outer automorphism of \(F_n\). Thus \(\Phi\in\operatorname{Aut}(F_n)\) represents \(\varphi\in\text{Out}(F_n)\).

Let \(R_n\) be the rose with one vertex \(*\) and \(n\) petals. A marked graph \(G\) is a graph of rank \(n\) all of whose vertices have valence at least two, equipped with a homotopy equivalence \(m\colon R_n\to G\). If \(b=m(*)\), it is obtained an identification of \(F_n\) with \(\pi_1(G,b)\). A homotopy equivalence \(f\colon G\to G\) and a path \(\sigma\) from \(b\) to \(f(b)\) determines an automorphism of \(\pi_1(G,b)\) and hence an automorphism \(\Phi\) of \(F_n\). This automorphism depends only on the homotopy class of \(\sigma\). So, modulo the inner automorphisms of \(F_n\), the \(f\colon G\to G\) represents \(\varphi\in\text{Out}(F_n)\).

Let \(G\) be a marked graph with marking \(m\colon R_n\to G\) and \(\Gamma\) its universal cover. The set \(\mathcal E(\Gamma)\) of ends of \(\Gamma\) is naturally identified with the boundary \(\partial F_n\) of \(F_n\). If \(f\colon G\to G\) represents \(\varphi\in\text{Out}(F_n)\), then a path \(\sigma\) from \(b\) to \(f(b)\) determines both an automorphism representing \(\varphi\) and a lift \(\widetilde f\colon\Gamma\to\Gamma\) of \(f\colon G\to G\). This defines a bijection between the set of lifts \(\widetilde f\) of \(f\) and the set of automorphisms \(\Phi\) of \(F_n\) representing \(\varphi\). Therefore \(\widetilde f\) is determined by \(\Phi\). Under the identification of \(\mathcal E(\Gamma)\) with the \(\partial F_n\) of \(F_n\), a lift \(\widetilde f\) determines a homeomorphism \(\widehat f\) of \(\partial F_n\). An automorphism \(\Phi\) also determines a homeomorphism \(\widehat\Phi\) of \(\partial F_n\) and \(\widehat f=\widehat\Phi\) if and only if \(\widetilde f\) is determined by \(\Phi\).

A point \(P\in\partial F_n\) is an attractor for \(\widehat\Phi\) if it has a neighborhood \(U\) such that \(\widehat\Phi(U)\subset U\) and \(\bigcap_{n=1}^\infty\widehat\Phi^n(U)=P\). If \(Q\) is an attractor for \(\widehat\Phi^{-1}\), then it is a repeller for \(\widehat\Phi\).

For the universal cover \(\Gamma\) of a marked graph \(G\) the space of lines in \(\Gamma\) is endowed with a weak topology. Then the graph \(G\) is equipped with the quotient topology. A closed set of lines in \(G\) or a closed \(F_n\)-invariant set of lines in \(\Gamma\) is called lamination. To each \(\varphi\in\text{Out}F_n)\) is associated a finite \(\varphi\)-invariant set of laminations \(\mathcal L(\varphi)\) called the set of attracting laminations for \(\varphi\).

The relative train track maps have been introduced by M. Bestvina and M. Handel [in Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)], in the present paper they play a central role and here the authors elaborate this notion and prove a relevant theorem (Theorem 2.19 in the paper).

Let \(f\colon G\to G\) be a relative train track map representing \(\varphi\in\text{Out}(F_n)\). For \(\Phi\in\operatorname{Aut}(F_n)\) representing \(\varphi\), let \(\text{Fix}_N(\widehat\Phi)\subset\text{Fix}(\widehat\Phi)\) be the set of non-repelling fixed points of \(\widehat\Phi\). \(\Phi\) is a principal automorphism if either of the following hold.

\(\bullet\) \(\text{Fix}_N(\widehat\Phi)\) contains at least three points.

\(\bullet\) \(\text{Fix}_N(\widehat\Phi)\) is a two point set that is neither the set of endpoints of an axis \(A_c\) nor the set of endpoints of a lift \(\widetilde\lambda\) of \(\mathcal L(\varphi)\). The corresponding lift \(\widetilde f\colon\Gamma\to\Gamma\) of \(f\) is a principal lift.

The set of principal automorphisms representing \(\varphi\) is denoted \(P(\varphi)\).

If \(\Phi\) is a principal lift of \(\varphi\), then \(\Phi^k\) is a principal lift of \(\varphi^k\) and \(\text{Fix}_N(\widehat\Phi)\subset\text{Fix}_N(\widehat\Phi^k)\) for every \(k\geq 1\). The set of all non-repelling periodic points in \(\text{Per}(\widehat\Phi)\) is denoted \(\text{Per}_N(\widehat\Phi)\). An outer automorphism \(\varphi\) is forward rotationless if \(\text{Fix}_N(\widehat\Phi)=\text{Per}_N(\widehat\Phi)\) for all \(\Phi\in P(\varphi)\) and if for each \(k\geq 1\) \(\Phi\mapsto\Phi^k\) defines a bijection between \(P(\varphi)\) and \(P(\varphi^k)\).

Now we are in the position to state the main Theorem of this paper.

Theorem: (Recognition Theorem) Suppose that \(\varphi,\psi\in\text{Out}(F_n)\) are forward rotationless and that

(1) \(PF_\Lambda(\varphi)=PF_\Lambda(\psi)\), for all \(\Lambda\in\mathcal L(\varphi)=\mathcal(\psi)\); and

(2) there is a bijection \(B\colon P(\varphi)\to P(\psi)\) such that:

(i) (fixed sets preserved) \(\text{Fix}_N(\widehat\Phi)=\text{Fix}_N(\widehat{B(\Phi)})\); and

(ii) (twist coordinates preserved) if \(w\in\text{Fix}(\Phi)\) and \(\Phi,i_w\Phi\in P(\varphi)\), then \(B(i_w\Phi)=i_wB(\Phi)\).

Then \(\varphi=\psi\).

Here \(i_w\) denotes the inner automorphism of \(F_n\) induced by \(w\) and, for each \(\Lambda\in\mathcal L(\varphi)\), \(PF_\Lambda\colon\text{Stab}(\Lambda)\to\mathbb Z\) is a homomorphism such that \(\psi\in\text{Ker}(PF_\Lambda)\) if and only if \(\Lambda\not\in\mathcal L(\psi)\) and \(\Lambda\not\in\mathcal L(\psi^{-1})\) (Lemma 2.14 in the paper).

The paper is based on previous articles, mainly on the previously mentioned one and on [M. Bestvina and the authors, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025)] and on [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, Duke Math. J. 93, No. 3, 425-452 (1998; Zbl 0946.20010)]. But here the authors go further in the study of some notions and develop new results.

Let \(R_n\) be the rose with one vertex \(*\) and \(n\) petals. A marked graph \(G\) is a graph of rank \(n\) all of whose vertices have valence at least two, equipped with a homotopy equivalence \(m\colon R_n\to G\). If \(b=m(*)\), it is obtained an identification of \(F_n\) with \(\pi_1(G,b)\). A homotopy equivalence \(f\colon G\to G\) and a path \(\sigma\) from \(b\) to \(f(b)\) determines an automorphism of \(\pi_1(G,b)\) and hence an automorphism \(\Phi\) of \(F_n\). This automorphism depends only on the homotopy class of \(\sigma\). So, modulo the inner automorphisms of \(F_n\), the \(f\colon G\to G\) represents \(\varphi\in\text{Out}(F_n)\).

Let \(G\) be a marked graph with marking \(m\colon R_n\to G\) and \(\Gamma\) its universal cover. The set \(\mathcal E(\Gamma)\) of ends of \(\Gamma\) is naturally identified with the boundary \(\partial F_n\) of \(F_n\). If \(f\colon G\to G\) represents \(\varphi\in\text{Out}(F_n)\), then a path \(\sigma\) from \(b\) to \(f(b)\) determines both an automorphism representing \(\varphi\) and a lift \(\widetilde f\colon\Gamma\to\Gamma\) of \(f\colon G\to G\). This defines a bijection between the set of lifts \(\widetilde f\) of \(f\) and the set of automorphisms \(\Phi\) of \(F_n\) representing \(\varphi\). Therefore \(\widetilde f\) is determined by \(\Phi\). Under the identification of \(\mathcal E(\Gamma)\) with the \(\partial F_n\) of \(F_n\), a lift \(\widetilde f\) determines a homeomorphism \(\widehat f\) of \(\partial F_n\). An automorphism \(\Phi\) also determines a homeomorphism \(\widehat\Phi\) of \(\partial F_n\) and \(\widehat f=\widehat\Phi\) if and only if \(\widetilde f\) is determined by \(\Phi\).

A point \(P\in\partial F_n\) is an attractor for \(\widehat\Phi\) if it has a neighborhood \(U\) such that \(\widehat\Phi(U)\subset U\) and \(\bigcap_{n=1}^\infty\widehat\Phi^n(U)=P\). If \(Q\) is an attractor for \(\widehat\Phi^{-1}\), then it is a repeller for \(\widehat\Phi\).

For the universal cover \(\Gamma\) of a marked graph \(G\) the space of lines in \(\Gamma\) is endowed with a weak topology. Then the graph \(G\) is equipped with the quotient topology. A closed set of lines in \(G\) or a closed \(F_n\)-invariant set of lines in \(\Gamma\) is called lamination. To each \(\varphi\in\text{Out}F_n)\) is associated a finite \(\varphi\)-invariant set of laminations \(\mathcal L(\varphi)\) called the set of attracting laminations for \(\varphi\).

The relative train track maps have been introduced by M. Bestvina and M. Handel [in Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)], in the present paper they play a central role and here the authors elaborate this notion and prove a relevant theorem (Theorem 2.19 in the paper).

Let \(f\colon G\to G\) be a relative train track map representing \(\varphi\in\text{Out}(F_n)\). For \(\Phi\in\operatorname{Aut}(F_n)\) representing \(\varphi\), let \(\text{Fix}_N(\widehat\Phi)\subset\text{Fix}(\widehat\Phi)\) be the set of non-repelling fixed points of \(\widehat\Phi\). \(\Phi\) is a principal automorphism if either of the following hold.

\(\bullet\) \(\text{Fix}_N(\widehat\Phi)\) contains at least three points.

\(\bullet\) \(\text{Fix}_N(\widehat\Phi)\) is a two point set that is neither the set of endpoints of an axis \(A_c\) nor the set of endpoints of a lift \(\widetilde\lambda\) of \(\mathcal L(\varphi)\). The corresponding lift \(\widetilde f\colon\Gamma\to\Gamma\) of \(f\) is a principal lift.

The set of principal automorphisms representing \(\varphi\) is denoted \(P(\varphi)\).

If \(\Phi\) is a principal lift of \(\varphi\), then \(\Phi^k\) is a principal lift of \(\varphi^k\) and \(\text{Fix}_N(\widehat\Phi)\subset\text{Fix}_N(\widehat\Phi^k)\) for every \(k\geq 1\). The set of all non-repelling periodic points in \(\text{Per}(\widehat\Phi)\) is denoted \(\text{Per}_N(\widehat\Phi)\). An outer automorphism \(\varphi\) is forward rotationless if \(\text{Fix}_N(\widehat\Phi)=\text{Per}_N(\widehat\Phi)\) for all \(\Phi\in P(\varphi)\) and if for each \(k\geq 1\) \(\Phi\mapsto\Phi^k\) defines a bijection between \(P(\varphi)\) and \(P(\varphi^k)\).

Now we are in the position to state the main Theorem of this paper.

Theorem: (Recognition Theorem) Suppose that \(\varphi,\psi\in\text{Out}(F_n)\) are forward rotationless and that

(1) \(PF_\Lambda(\varphi)=PF_\Lambda(\psi)\), for all \(\Lambda\in\mathcal L(\varphi)=\mathcal(\psi)\); and

(2) there is a bijection \(B\colon P(\varphi)\to P(\psi)\) such that:

(i) (fixed sets preserved) \(\text{Fix}_N(\widehat\Phi)=\text{Fix}_N(\widehat{B(\Phi)})\); and

(ii) (twist coordinates preserved) if \(w\in\text{Fix}(\Phi)\) and \(\Phi,i_w\Phi\in P(\varphi)\), then \(B(i_w\Phi)=i_wB(\Phi)\).

Then \(\varphi=\psi\).

Here \(i_w\) denotes the inner automorphism of \(F_n\) induced by \(w\) and, for each \(\Lambda\in\mathcal L(\varphi)\), \(PF_\Lambda\colon\text{Stab}(\Lambda)\to\mathbb Z\) is a homomorphism such that \(\psi\in\text{Ker}(PF_\Lambda)\) if and only if \(\Lambda\not\in\mathcal L(\psi)\) and \(\Lambda\not\in\mathcal L(\psi^{-1})\) (Lemma 2.14 in the paper).

The paper is based on previous articles, mainly on the previously mentioned one and on [M. Bestvina and the authors, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025)] and on [D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig, Duke Math. J. 93, No. 3, 425-452 (1998; Zbl 0946.20010)]. But here the authors go further in the study of some notions and develop new results.

Reviewer: Dimitrios Varsos (Athenai)

##### MSC:

20E36 | Automorphisms of infinite groups |

20E05 | Free nonabelian groups |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics |

##### Keywords:

outer automorphisms; free groups; train tracks; attracting laminations; outer automorphism groups##### References:

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