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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In unpublished notes (1998) it is proved, by G. Levitt and M. Lustig, that \(e\leq\tfrac{3n-2}{4}\).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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