The inclusion of the simplicial boundary of Outer space into its 2-bond thickening is (2n-3)-connected, with all non-contractible fibres concentrated over theta graphs.
Relative free splitting and free factor complexes I: Hyperbolicity
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We study the large scale geometry of the relative free splitting complex and the relative free factor complex of the rank $n$ free group $F_n$, relative to the choice of a free factor system of $F_n$, proving that these complexes are hyperbolic. Furthermore we present the proof in a general context, obtaining hyperbolicity of the relative free splitting complex and of the relative free factor complex of a general group $\Gamma$, relative to the choice of a free factor system of $\Gamma$. The proof yields information about coarsely transitive families of quasigeodesics in each of these complexes, expressed in terms of fold paths of free splittings.
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Bond thickenings of the simplicial boundary of Outer space
The inclusion of the simplicial boundary of Outer space into its 2-bond thickening is (2n-3)-connected, with all non-contractible fibres concentrated over theta graphs.