Given a finite rank free group F N 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 = ψ m , φ n 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.
Given a finitely generated free group
$ {\mathbb {F} }$
of
$\mathsf {rank}( {\mathbb {F} } )\geq 3$
, we show that the mapping torus of
$\phi$
is (strongly) relatively hyperbolic if
$\phi$
is exponentially growing. As a corollary of our work, we give a new proof of Brinkmann's theorem which proves that the mapping torus of an atoroidal outer automorphism is hyperbolic. We also give a new proof of the Bridson–Groves theorem that the mapping torus of a free group automorphism satisfies the quadratic isoperimetric inequality. Our work also solves a problem posed by Minasyan and Osin: the mapping torus of an outer automorphism is not virtually acylindrically hyperbolic if and only if
$\phi$
has finite order.
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