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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In unpublished notes (1998) it is proved, by G. Levitt and M. Lustig, that \(e\leq\tfrac{3n-2}{4}\).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 \(d=0\)).

20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F65 Geometric group theory
20E08 Groups acting on trees
37B10 Symbolic dynamics
Full Text: DOI arXiv
[1] Bestvina M., Feighn M., Handel M.: The Tits alternative for Out(F n ), I, Dynamics of exponentially-growing automorphisms. Ann. of Math. 151, 517–623 (2000) · Zbl 0984.20025
[2] Bestvina M., Feighn M., Handel M.: The Tits alternative for Out(F n ), II, A Kolchin type theorem. Ann. of Math. 161, 1–59 (2005) · Zbl 1139.20026
[3] Bestvina M., Handel M.: Train tracks for automorphisms of the free group. Ann. Math. 135, 1–51 (1992) · Zbl 0757.57004
[4] M. Bridson, D. Groves, The quadratic isoperimetric inequality for mapping tori of free group automorphisms, Memoirs of the AMS, to appear. · Zbl 1201.20037
[5] Collins D.J., Turner E.C.: All automorphisms of free groups with maximal rank fixed subgroups. Math. Proc. Camb. Phil. Soc. 119, 615–630 (1996) · Zbl 0849.20018
[6] Dyer J.L., Scott P.G.: Periodic automorphisms of free groups. Comm. Algebra 3, 195–201 (1975) · Zbl 0304.20029
[7] M. Handel, L. Mosher, Subgroup classification in Out(F n ), arXiv:0908.1255. · Zbl 1285.20033
[8] Gaboriau D., Jaeger A., Levitt G., Lustig M.: An index for counting fixed points of automorphisms of free groups. Duke Math. Jour. 93, 425–452 (1998) · Zbl 0946.20010
[9] Gaboriau D., Levitt G.: The rank of actions on R-trees. Ann. Sc. ENS 28, 549–570 (1995) · Zbl 0835.20038
[10] Gaboriau D., Levitt G., Lustig M.: A dendrological proof of the Scott conjecture for automorphisms of free groups. Proc. Edinburgh Math. Soc. 41, 325–332 (1998) · Zbl 0904.20014
[11] F. Gautero, M. Lustig, The mapping-torus of a free group automorphism is hyperbolic relative to the canonical subgroups of polynomial growth, arXiv:0707.0822.
[12] G. Levitt, M. Lustig, Unpublished notes (1998).
[13] Levitt G., Lustig M.: Periodic ends, growth rates, Hölder dynamics for automorphisms of free groups. Comm. Math. Helv. 75, 415–430 (2000) · Zbl 0965.20026
[14] A. Piggott, Detecting the growth of free group automorphisms by their action on the homology of subgroups of finite index, math.GR/0409319.
[15] H. Short, Quasiconvexity and a theorem of Howson’s, in ”Group Theory from a Geometrical Viewpoint” (Ghys, Haefliger, Verjovsky, eds.),World Scientific (1991), 168–176. · Zbl 0869.20023
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