A Cut Vertex Lemma

10 May 2022

Building on ideas from the previous post regarding the star graph, I wrote a short paper generalizing a classical lemma of Whitehead. The purpose of this post is to describe Whitehead’s lemma and my generalization.

In the previous post, we met the star graph associated to a collection of conjugacy classes of elements of a virtually free group. It should be reasonably clear that one can extend the notion of a star graph to conjugacy classes of elements of a group that acts cocompactly on a tree. What is perhaps less obvious is that one can also extend this notion to convex-cocompact subgroups of groups acting on trees, where a subgroup is convex-cocompact if it acts cocompactly on its minimal subtree. This generalization is due to Bestvina, Feighn and Handel in the case of free groups.

The idea is the following. A classical observation of Stallings says that any finitely generated subgroup \(H\) of a free group \(G\) may be represented by an immersion of a finite graph \(\Gamma\) into a rose \(R\), that is a graph with one vertex \(w\) and \(n\) edges, where \(n\) is the rank of \(G\). The set of directions \(D_w\) is the set of oriented edges of \(R\). Cut each edge of \(\Gamma\) at its midpoint, observe that the half-edge left over crosses a direction of \(R\) and attach, in the star graph of \(\Gamma\), the half-edge to the vertex corresponding to the given direction in \(D_w\). Thus we get a bipartite graph with one vertex for each vertex of \(\Gamma\) and one vertex for each direction in \(D_w\). A pair of directions are connected by a path of length two in this star graph if and only if the corresponding turn is in the image of the map on turns associated to the immersion \(\Gamma \to R\).

This idea generalizes readily to \(G\) the fundamental group of a graph of groups \(\mathbb{G}\). A subgroup of \(G\) is convex-cocompact if and only if it may be represented as an immersion of a finite connected graph of groups \(f\colon \mathcal{G} \to \mathbb{G}\). The set of directions we have met before. Rather than snipping edges at midpoints in \(\mathcal{G}\), we essentially do this in the Bass–Serre tree. If \(v\) is a vertex of \(\mathcal{G}\) mapping to a vertex \(w\) in \(\mathbb{G}\), we attach \(|\mathbb{G}_w : f_v(\mathcal{G}_v)|\) copies of this piece of the Bass–Serre tree, where the \(gf_v(\mathcal{G}_v)\) copy attaches an edge corresponding to a direction \(d\) in \(D_v\) to the vertex corresponding to the direction \(g.D_vf(d)\) in \(D_w\).

Suppose there exists \(\mathcal{G}'\) a graph of groups with fundamental group \(G\) and one edge with the property that \(\mathcal{G}'\) may be obtained from a graph of groups homotopy equivalent to \(\mathbb{G}\) by collapsing edges. Anyway, say that a conjugacy class of an element or convex-cocompact subgroup of \(G\) is simple if there is such a graph of groups \(\mathcal{G}'\) with the property that our element or subgroup is conjugate into a vertex group in \(\mathcal{G}'\). For free groups thought of as fundamental groups of ordinary graphs, it’s not hard to see that an element or finitely generated (hence convex-cocompact) subgroup is simple if and only if it is contained in a proper free factor of the free group.

The main result of my paper is that if we have a jointly simple collection \(C\) of conjugacy classes of elements or convex-cocompact subgroups of \(G\) (i.e. the same \(\mathcal{G}'\) works for all members of \(C\) at once) then the star graph of \(C\) is either disconnected or has a cut vertex.

“Ahh but Rylee,” you say astutely, paying far too much attention, “Isn’t the star graph already disconnected if \(\mathbb{G}\) has more than one vertex?” To which I have to concede that you are correct and explain my usage further. The star graph is a union of graphs \(\Gamma_w\) as \(w\) varies over the vertices of \(\mathbb{G}\), where \(\Gamma_w\) contains all the vertices corresponding to directions in \(D_w\). We say that the star graph is disconnected if some \(\Gamma_w\) is disconnected, and that it has a cut vertex if each \(\Gamma_w\) is connected and there is some direction \(d \in D_w\), the removal of which disconnects \(\Gamma_w\).

The proof is very much inspired by a paper of Heusener and Weidmann, which gives a proof of the classical Whitehead lemma for free groups using Stallings folds. The idea is that if a collection of conjugacy classes is simple, there is a graph which folds onto the rose with a single fold (their term is “almost rose”) in which these conjugacy classes may be read, in the sense that the immersion \(f\colon \Gamma \to R\) factors through the almost rose. They prove that (their version of) the star graph of an almost rose has a cut vertex and that if the map \(f\colon \Gamma \to R\) factors through a graph \(\Theta\), then the star graph of \(\Gamma\) is a subgraph of the star graph of \(\Theta\). Although we are working with a different notion of the star graph, the broad-strokes outline of the proof carries over to our graph of groups setting! If you’d like to know more, you can read the paper here.