Applications of weak attraction theory in $\mathrm {Out}(\mathbb {F}_N)$

Abstract

Given a finite rank free group $\mathbb {F }_N$ of rank $\ge 3$ and two exponentially growing outer automorphisms $\psi$ and $\phi$ with dual lamination pairs ${\varLambda }^\pm _\psi$ and ${\varLambda }^\pm _\phi$ associated to them, which satisfy a notion of independence described in this paper, we will use the pingpong techniques developed by Handel and Mosher (Subgroup decomposition in Out(F_n), part III: weak attraction theory, 2013) to show that there exists an integer $M > 0$, such that for every $m,n\ge M$, the group $G=\langle \psi ^m,\phi ^n \rangle$ will be a free group of rank two and every element of this free group which is not conjugate to a power of the generators will be fully irreducible and hyperbolic.