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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.

##### MSC:
 20E05 Free nonabelian groups 20E36 Automorphisms of infinite groups 20F65 Geometric group theory
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##### References:
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