Index realization for automorphisms of free groups

TL;DR

This paper extends a surface homeomorphism result to automorphisms of free groups, establishing an index inequality analogous to the Gauss-Bonnet relation, for fully irreducible automorphisms.

Contribution

It proves an analogue of a surface foliation result for free group automorphisms, replacing the Gauss-Bonnet equality with an index inequality.

Findings

Establishes the index inequality for fully irreducible automorphisms of free groups.

Provides a construction method for automorphisms with prescribed index properties.

Bridges concepts between surface homeomorphisms and free group automorphisms.

Abstract

For any surface $\Sigma$ of genus $g \geq 1$ and (essentially) any collection of positive integers $i_1, i_2, \ldots, i_\ell$ with $i_1+\cdots +i_\ell = 4g-4$ Masur and Smillie 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 $\cal L$, such that $\cal L$ 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 $i_1+\cdots +i_\ell \leq 2N-2$ from Gaboriau, Jaeger, Levitt and Lustig.