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The expansion factors of an outer automorphism and its inverse. (English) Zbl 1127.20021
Let \(F_n\) be the free group of rank \(n\). An outer automorphism \(\varphi\) is ‘fully irreducible’ if there are no free factors of \(F_n\) that are invariant under an iterate of \(\varphi\). A fully irreducible outer automorphism \(\varphi\) has an expansion factor that can be thought of as the exponential growth rate of the action of \(\varphi\) on the conjugacy classes in \(F_n\). In general the expansion factor, say \(\lambda\), of \(\varphi\) is not equal to the expansion factor of \(\varphi^{-1}\), say \(\mu\) (In [Trans. Am. Math. Soc. 359, No. 7, 3153-3183 (2007; Zbl 1120.20042)] the authors exhibit the property of a fully irreducible outer automorphism to have different expansion factor from its inverse.)
The aim of this work is to measure the variation between \(\lambda\) and \(\mu\). More exactly it is proved the following Theorem. Suppose that \(\varphi\in\text{Out}(F_n)\) is fully irreducible, that \(\lambda\) is the expansion factor of \(\varphi\) and that \(\mu\) is the expansion factor of \(\varphi^{-1}\). Then \(\log(\lambda)\sim\log(\mu)\).
Here \(\log(\lambda)\sim\log(\mu)\) means that the ratio \(\log(\lambda)/\log(\mu)\) is bounded above and below ‘uniformly’, namely these bounds are not dependent upon \(\varphi\) but only on the free rank \(n\) of \(F_n\).
For an arbitrary outer automorphism \(\varphi\) of \(F_n\) it is defined a finite set of expansion factors of \(\varphi\). There is a bijective pairing between the expansion factors of \(\varphi\) and its inverse \(\varphi^{-1}\).
A generalization of the above Theorem is obtained. Given \(\varphi\in\text{Out}(F_n)\), if \((\lambda_i),(\mu_i)\) are the paired expansion factors of \(\varphi\), \(\varphi ^{-1}\), respectively, then \(\log(\lambda_i)\sim\log(\mu_i)\) for all \(i\).
Although for the proofs of the above theorems and relevant definitions the authors put on [M. Bestvina, M. Handel, Ann. Math. (2) 135, No. 1, 1-51 (1992; Zbl 0757.57004)] and M. Bestvina, M. Feighn, M. Handel, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025)] the paper is self contained, but if someone is interested not only in the results, he should first study the above mentioned papers.

20E05 Free nonabelian groups
20E36 Automorphisms of infinite groups
20F65 Geometric group theory
Full Text: DOI arXiv
[1] Emina Alibegović, Translation lengths in \?\?\?(\?_\?), Geom. Dedicata 92 (2002), 87 – 93. Dedicated to John Stallings on the occasion of his 65th birthday. · Zbl 1041.20024 · doi:10.1023/A:1019695003668 · doi.org
[2] Mladen Bestvina, Mark Feighn, and Michael Handel, The Tits alternative for \?\?\?(\?_\?). I. Dynamics of exponentially-growing automorphisms, Ann. of Math. (2) 151 (2000), no. 2, 517 – 623. · Zbl 0984.20025 · doi:10.2307/121043 · doi.org
[3] Mladen Bestvina and Michael Handel, Train tracks and automorphisms of free groups, Ann. of Math. (2) 135 (1992), no. 1, 1 – 51. · Zbl 0757.57004 · doi:10.2307/2946562 · doi.org
[4] M. Handel and L. Mosher, Parageometric outer automorphisms of free groups, Trans. Amer. Math. Soc., this issue. · Zbl 1120.20042
[5] Gilbert Levitt and Martin Lustig, Irreducible automorphisms of \?_\? have north-south dynamics on compactified outer space, J. Inst. Math. Jussieu 2 (2003), no. 1, 59 – 72. · Zbl 1034.20038 · doi:10.1017/S1474748003000033 · doi.org
[6] John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551 – 565. , https://doi.org/10.1007/BF02095993 S. M. Gersten, Intersections of finitely generated subgroups of free groups and resolutions of graphs, Invent. Math. 71 (1983), no. 3, 567 – 591. · Zbl 0521.20014 · doi:10.1007/BF02095994 · doi.org
[7] Karen Vogtmann, Automorphisms of free groups and outer space, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 1 – 31. · Zbl 1017.20035 · doi:10.1023/A:1020973910646 · doi.org
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