The free splitting complex of a free group I: Hyperbolicity

Abstract

We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.

85 pages, including index and glossary. Many figures in the free splitting complex, in the guise of commutative diagrams of maps between free splittings. Changes from Version 1 to Version 2: Corrected an error in the proof of Proposition 6.5, Step 2. Revamped the theory of free splitting units in Section 5

References in corpus (6)

• Geometry of the Complex of Curves I: Hyperbolicity
• Core and intersection number for group actions on trees
• Geometry of pseudocharacters
• On train track splitting sequences
• Lipschitz retraction and distortion for subgroups of Out(F_n)
• On the geometry of a proposed curve complex analogue for $Out(F_n)$

Cited by in corpus (37)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• Uniform hyperbolicity of the curve graph via surgery sequences
• The boundary of the complex of free factors
• Geometry and dynamics in Gromov hyperbolic metric spaces: With an emphasis on non-proper settings
• Lines of minima in Outer space
• Constructing group actions on quasi-trees and applications to mapping class groups
• The boundary of the free splitting graph and the free factor graph
• Subfactor projections
• The complex of partial bases for F_n and finite generation of the Torelli subgroup of Aut(F_n)
• Statistics and compression of scl
• Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
• Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Relative free splitting and free factor complexes I: Hyperbolicity
• North-South dynamics of hyperbolic free group automorphisms on the space of currents
• A note on subfactor projections
• Right-angled Artin groups and Out(F_n) I: quasi-isometric embeddings
• Submanifold Projection
• Automorphisms of graphs of cyclic splittings of free groups
• Canonical measures on metric graphs and a Kazhdan's theorem
• Random walks, WPD actions, and the Cremona group
• Spheres and Projections for $\mathrm{Out}(F_n)$
• The mapping class group of connect sums of $S^2\times S^1$
• Hyperbolicity of the Cyclic Splitting Complex
• Hyperbolic actions and 2nd bounded cohomology of subgroups of $\text{Out}(F_n)$. Part I: Infinite lamination subgroups
• Atoroidal dynamics of subgroups of Out(F_N)
• Uniform fellow traveling between surgery paths in the sphere graph
• Hyperelliptic graphs and the period mapping on outer space
• Splittings of free groups from arcs and curves
• The Pants Graph of a Free Group
• Spotted disk and sphere graphs I
• Weak tight geodesics in the curve complex: examples and gaps
• Cocycle superrigidity from higher rank lattices to $\mathrm{Out}(F_N)$
• The minimally displaced set of an irreducible automorphism is locally finite
• The Complex of Non-separating embedded spheres
• Quasi-geodesics in Out(F_n) and their shadows in sub-factors
• Loxodromic elements in the cyclic splitting complex and their centralizers