Illinois Journal of Mathematics

Index realization for automorphisms of free groups

Thierry Coulbois and Martin Lustig

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For any surface $\Sigma$ of genus $g\geq1$ and (essentially) any collection of positive integers $i_{1},i_{2},\ldots,i_{\ell}$ with $i_{1}+\cdots+i_{\ell}=4g-4$ Masur and Smillie (Comment. Math. Helv. 68 (1993) 289–307) have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma\to\Sigma$ 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 (Duke Math. J. 93 (1998) 425–452).

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 1111-1128.

Received: 2 May 2016
Revised: 5 December 2016
First available in Project Euclid: 27 February 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20E05: Free nonabelian groups 20E08: Groups acting on trees [See also 20F65] 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 57R30: Foliations; geometric theory


Coulbois, Thierry; Lustig, Martin. Index realization for automorphisms of free groups. Illinois J. Math. 59 (2015), no. 4, 1111--1128. doi:10.1215/ijm/1488186023.

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