Acylindrical Hyperbolicity of $\mathrm{Out}(W_n)$
Keywords:
acylindrical hyperbolicity, automorphisms, Coxeter group, fully irreducible automorphisms, generalized loxodromic, outer automorphismsAbstract
We prove that the group of outer automorphisms of the free Coxeter group $W_n$ is acylindrically hyperbolic in the sense of Osin. We observe that any $\mathrm{CAT}(0)$ space admitting a geometric action by $\mathrm{Out}(W_n)$ must contain a rank-one geodesic as an application. Our main theorem proceeds from expanding on a well-known relationship between $\mathrm{Out}(W_n)$ and the outer automorphism group of free groups.
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