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Relative free splitting and free factor complexes I: Hyperbolicity

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arxiv 1407.3508 v4 pith:VBJPHCR5 submitted 2014-07-13 math.GR

classification math.GR
keywords freerelativefactorcomplexcomplexessplittingchoicegamma
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abstract

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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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Bond thickenings of the simplicial boundary of Outer space

    math.AT 2026-08 accept novelty 8.0 of 10

    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.

  2. Understanding the well-rounded deformation retraction of Teichm\"uller space

    math.GT 2025-09 conditional novelty 6.0 of 10

    The author proves, modulo two of her own unpublished preprints, that Teichmüller space has a well-rounded, equivariant deformation retraction onto a complex of dimension 4g-5.

  3. Relative free splitting and free factor complexes: An overview

    math.GR 2026-07 conditional novelty 2.0 of 10

    An overview of new theorems on the hyperbolicity and geometric dynamics of relative free splitting and free factor complexes, with proofs deferred to three companion papers by the same authors.

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