A Cut Vertex Lemma

May 10, 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.