Distance in the Ellipticity Graph

Abstract

The ellipticity graph of a free group $F$ was defined by I. Kapovich and M. Lustig in order to study the outer automorphism group of $F$, which acts on this graph. The graph was constructed to be analogous to the curve complex of a surface. It is a bipartite graph, whose vertices are conjugacy classes of nontrivial elements of $F$ and equivalence classes of proper free product decompositions of the form $F=A\ast B$. A conjugacy class is joined by an edge to a free product decomposition $A\ast B$ whenever the conjugacy class has a representative in either $A$ or $B$. This paper uses Stallings subgroup $X$-digraphs and Whitehead automorphisms to construct algorithms that determine when the distance between two vertices of the ellipticity graph is two, for both types of vertices.

The author was informed during the review process that these results have been previously known. This revised version provides those references, along with a broader introduction to the topic. This version also omits the more minor results in section 5 of the previous version

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