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Relative hyperbolicity of free-by-cyclic extensions
- Part of
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- Journal:
- Compositio Mathematica / Volume 159 / Issue 1 / January 2023
- Published online by Cambridge University Press:
- 30 January 2023, pp. 153-183
- Print publication:
- January 2023
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- Article
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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.
