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

TL;DR

This paper characterizes the Gromov boundaries of the free factor, free splitting, and cyclic splitting graphs for free groups, describing them in terms of equivalence classes of certain minimal, indecomposable trees with specific properties.

Contribution

It provides a detailed description of the Gromov boundaries of key graphs associated with free groups, linking them to classes of minimal, very small, indecomposable trees.

Findings

Boundary of the free factor graph is the space of equivalence classes of minimal very small indecomposable trees.

Boundary of the free splitting graph includes trees that are either indecomposable or split as large graphs of actions.

Boundary of the cyclic splitting graph consists of trees that are either indecomposable or split as very large graphs of actions.

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.