The expansion factors of an outer automorphism and its inverse.

*(English)*Zbl 1127.20021Let \(F_n\) 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.

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.

Reviewer: Dimitrios Varsos (Athenai)

##### MSC:

20E05 | Free nonabelian groups |

20E36 | Automorphisms of infinite groups |

20F65 | Geometric group theory |

##### References:

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