Index realization for automorphisms of free groups

Abstract

For any surface $\Sigma$ of genus $g\ge 1$ and (essentially) any collection of positive integers $i_1,i_2,\dots ,i_{\ell }$ with $i_1+\cdots +i_{\ell }=4g-4$ Masur and Smillie have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma \to \Sigma$ with precisely $\ell$ singularities $S_1,\dots ,S_{\ell }$ 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 $i_1+\cdots +i_{\ell }\le 2N-2$ from Gaboriau, Jaeger, Levitt and Lustig.

arXiv comment19 pages, 3 figures, 1 table, revised version for publication