The boundary of the free splitting graph and the free factor graph

Abstract

We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the union of their metric completions with their Gromov boundaries are Fn-equivariantly homeomorphic with respect to the observer's topology. The boundary of the cyclic splitting graph is the space of equivalence classes of trees which either are indecomposable or split as very large graph of actions. The boundary of the free splitting graph is the space of equivalence classes of trees which either are indecomposable or split as large graph of actions.

Argument clarified and streamlined. Details added. 60p

References in corpus (4)

• The boundary of the complex of free factors
• Reducing systems for very small trees
• Dynamics of Irreducible Endomorphisms of $F_n$
• Hyperbolicity of the Cyclic Splitting Complex

Cited by in corpus (14)

• The co-surface graph and the geometry of hyperbolic free group extensions
• Central limit theorems for mapping class groups and $\text{Out}(F_N)$
• Boundaries of relative factor graphs and subgroup classification for automorphisms of free products
• North-South dynamics of hyperbolic free group automorphisms on the space of currents
• Ergodic decompositions for folding and unfolding paths in Outer space
• A short proof of Handel and Mosher's alternative for subgroups of $\text{Out}(F_N)$
• Automorphisms of graphs of cyclic splittings of free groups
• Spectral theorems for random walks on mapping class groups and $\text{Out}(F_N)$
• Trees, dendrites, and the Cannon-Thurston map
• Random trees in the boundary of Outer space
• Long turns, INP's and index for free group automorphisms
• Measure equivalence rigidity of $\mathrm{Out}(F_N)$
• Commensurations of subgroups of $\mathrm{Out}(F_N)$
• Limit sets of unfolding paths in Outer space