On hyperbolicity of free splitting and free factor complexes

Abstract

We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map $\tau$ from the free splitting complex to free factor complex, a geodesic $[x,y]$ maps to a path that is uniformly Hausdorff-close to a geodesic $[\tau (x),\tau (y)]$ .

updated version, revised following the referee's comments; included a figure provided by the referee

Cited by in corpus (15)

• The boundary of the complex of free factors
• The co-surface graph and the geometry of hyperbolic free group extensions
• The Poisson boundary of $\text{Out}(F_N)$
• Hyperbolic graphs for free products, and the Gromov boundary of the graph of cyclic splittings
• Dynamics on free-by-cyclic groups
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• Counting loxodromics for hyperbolic actions
• Acylindrically hyperbolic groups with exotic properties
• Acylindrical group actions on quasi-trees
• Automorphisms of graphs of cyclic splittings of free groups
• The complex of partial bases of a free group
• The compression body graph has infinite diameter
• On purely loxodromic actions
• Quasi-geodesics in Out(F_n) and their shadows in sub-factors
• The Pants Graph of a Free Group