# 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.