Relative free splitting and free factor complexes I: Hyperbolicity

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.

86 pages + references. In the latest version, there are several small changes in support of the Overview (arXiv:2607.19249), of Part II (arXiv:2212.09907), and of Part III (arXiv:2503.07532)

References in corpus (3)

• The free splitting complex of a free group I: Hyperbolicity
• Constructing group actions on quasi-trees and applications to mapping class groups
• Subfactor projections

Cited by in corpus (9)

• The Tits alternative for the automorphism group of a free product
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Homotopy type of the complex of free factors of a free group
• Simultaneous construction of hyperbolic isometries
• Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups
• Irreducible laminations for IWIP Automorphisms of free products and Centralisers
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Commensurations of subgroups of $\mathrm{Out}(F_N)$