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The recognition theorem for \(\text{Out}(F_n)\). (English) Zbl 1239.20036
Let \(F_n\) 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.

20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
Full Text: DOI arXiv
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