## Illinois Journal of Mathematics

### Index realization for automorphisms of free groups

#### Abstract

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

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

Dates
Revised: 5 December 2016
First available in Project Euclid: 27 February 2017

https://projecteuclid.org/euclid.ijm/1488186023

Digital Object Identifier
doi:10.1215/ijm/1488186023

Mathematical Reviews number (MathSciNet)
MR3628303

Zentralblatt MATH identifier
1382.20030

#### Citation

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. https://projecteuclid.org/euclid.ijm/1488186023

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