Applications of weak attraction theory in Out(𝔽𝑁)

Abstract

Given a finite rank free group 𝔽𝑁 of rank ≥3 and two exponentially growing outer automorphisms 𝜓 and 𝜙 with dual lamination pairs 𝛬±𝜓 and 𝛬±𝜙 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≥𝑀, the group 𝐺=⟨𝜓𝑚,𝜙𝑛⟩ 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.