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Irreducible automorphisms of growth rate one. (English) Zbl 0787.20023

As the title indicates the authors study the irreducible automorphisms (of a finitely generated free group \(F\)) of growth rate one. This terminology comes from the known paper of M. Bestvina and M. Handel [Ann. Math., II. Ser. 135, 1-51 (1992; Zbl 0757.57004)]. Here the authors describe explicitly the irreducible automorphisms of growth rate one. They define an automorphism \(\alpha \in \text{Aut }F\) to be reducible if there exists a free decomposition \(F = (*_{i\in \mathbb{Z}_ m}F_ i) * F_ \infty\), for some \(m\geq 1\), such that for every \(i \in \mathbb{Z}_ m\) there exists an \(f_ i \in F\) such that \(F_ i^{\alpha f_ i} = F_{i+1}\) (\(\mathbb{Z}_ m\) is the set of integers \(\text{mod }m\)). An automorphism is called irreducible if it is not reducible.
The main result of this paper describes precisely the irreducible automorphisms of growth rate one: let \(p < q\) be prime numbers. Then the free group of rank \(pq-p-q+1\) can be presented as \(\langle x_{i,j} (i\in \mathbb{Z}_ p,\;j\in\mathbb{Z}_ q)\mid x_{i,0} = x_{0,j} = 1\rangle\), and the mapping \(x_{i,j}\mapsto x_{1,1} x^{- 1}_{i+1,1}x_{i+1,j+1}x^{-1}_{1,j+1}\), \(i\in \mathbb{Z}_ p\), \(j\in \mathbb{Z}_ q\) defines an automorphism denoted by \(\alpha_{pq}\). Let \(_ 0\) and \(_ 1\) denote the identity automorphisms of groups of rank 0 and 1, respectively and let \(\alpha_ 2\) denote the inverting automorphism of the free group of rank 1. They prove that an irreducible automorphism of a finitely generated free group with growth rate 1 is equivalent (in a certain sense) to one of the above automorphisms \(\alpha_ n\). They also study the irreducibility of the outer automorphisms as well as geometric realizations of the above irreducible automorphisms.

MSC:

20E36 Automorphisms of infinite groups
20E05 Free nonabelian groups
20F65 Geometric group theory
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations

Citations:

Zbl 0757.57004
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References:

[1] Bestvina, M.; Handel, M., Train tracks and automorphisms of free groups, Ann. of Math., 135, 1-51 (1992) · Zbl 0757.57004
[2] Casson, A. J.; Bleiler, S. A., Automorphisms of Surfaces after Nielsen and Thurston (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0649.57008
[3] Gersten, S. M.; Stallings, J. R., Irreducible outer automorphisms of a free groups, Proc. Amer. Math. Soc., 111, 309-314 (1991) · Zbl 0717.20026
[4] van der Waerden, B. L., Modern Algebra, Vol. I (1964), Frederick Ungar: Frederick Ungar New York · Zbl 0033.10102
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