Index realization for automorphisms of free groups. (English) Zbl 1382.20030

Coulbois, Thierry; Lustig, Martin

Ill. J. Math. 59, No. 4, 1111-1128 (2015).

Summary: For any surface \(\Sigma\) of genus \(g\geq 1\) and (essentially) any collection of positive integers \(i_{1},i_{2},\ldots,i_{\ell}\) with \(i_{1}+\cdots+i_{\ell}=4g-4\) H. Masur and J. Smillie [Comment. Math. Helv. 68, No. 2, 289–307 (1993; Zbl 0792.30030)] have shown that there exists a pseudo-Anosov homeomorphism \(h:\Sigma\rightarrow\Sigma\) with precisely \(\ell\) singularities \(S_{1},\ldots,S_{\ell}\) in its stable foliation \(\mathcal{L}\), such that \(\mathcal{L}\) has precisely \(i_{k}+2\) separatrices raying out from each \(S_{k}\). In this paper, we prove the analogue of this result for automorphisms of a free group \({F}_{N}\), where “pseudo-Anosov homeomorphism” is replaced by “fully irreducible automorphism” and the Gauss-Bonnet equality \(i_{1}+\cdots+i_{\ell}=4g-4\) is replaced by the index inequality \(i_{1}+\cdots+i_{\ell}\leq2N-2\) from [D. Gaboriau et al., Duke Math. J. 93, No. 3, 425–452 (1998; Zbl 0946.20010)].

MSC:

20E05	Free nonabelian groups
20E36	Automorphisms of infinite groups
20E08	Groups acting on trees
20F65	Geometric group theory
57R30	Foliations in differential topology; geometric theory

Keywords:

fully irreducible automorphism; index inequality; automorphisms of free groups

Citations:

Zbl 0792.30030; Zbl 0946.20010