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Counting growth types of automorphisms of free groups. (English) Zbl 1196.20038
Let $$F_n$$ be the free group of rank $$n$$. Given an automorphism $$\alpha$$ of $$F_n$$ three invariants are considered.
Let $$\Phi\in\text{Out}(F_n)$$ be the outer automorphism represented by $$\alpha$$, there exists $$q\geq 1$$ such that $$\Phi^q$$ is represented by an improved relative train track map [see M. Bestvina, M. Feighn, M. Handel, Ann. Math. (2) 151, No. 2, 517-623 (2000; Zbl 0984.20025), Th.5.1.5]. $$e$$ is the number of exponential strata of any improved relative train track map representing a power of $$\Phi$$. It is also the number of attracting laminations of $$\Phi$$ [loc. cit., §3.1]. $$d$$ is the maximal degree of polynomial growth of conjugacy classes. Namely it is the maximal degree such that the length of some conjugacy class grows as a polynomial of degree $$d$$ under iteration of $$\alpha$$. $$R$$ is the rank of the fixed subgroup of $$\alpha$$.
In this paper the author characterizes the triple $$(e,d,R)$$ for an automorphism of $$F_n$$.
Theorem 1. Given $$\alpha\in\operatorname{Aut}(F_n)$$, the numbers $$e$$ and $$d$$ satisfy: $$e+d\leq n-1$$, $$4e+2d\leq 3n-2$$ ($$\leq 3n-3$$ if $$d>0$$). Conversely, any $$(e,d)$$ satisfying these inequalities may be realized by some $$\alpha\in\operatorname{Aut}(F_n)$$.
In unpublished notes (1998) it is proved, by G. Levitt and M. Lustig, that $$e\leq\tfrac{3n-2}{4}$$.
Theorem 2. Given $$e$$ and $$d$$ satisfying the conditions above, the possible values of $$R=\text{rk\,Fix\,}\alpha$$ for an automorphism $$\alpha$$ of $$F_n$$ are exactly those allowed by the following inequalities: $$e+\max(d-1,0)+R\leq n$$, $$4e+2d+2R\leq 3n+1$$ ($$\leq 3n$$ if $$d=0$$).

##### MSC:
 20E36 Automorphisms of infinite groups 20E05 Free nonabelian groups 20F65 Geometric group theory 20E08 Groups acting on trees 37B10 Symbolic dynamics
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