In an earlier post we introduced the concept of an ideal edge in a graph of groups and described how to blow up ideal edges to construct new graphs of groups. To prove that (the reduced spine of) Outer Space for a virtually free group is contractible, Krstić and Vogtmann define what is essentially a combinatorial analogue of a non-singular Morse function on Outer Space and show that for a well-chosen function, the set of minimal vertices of the complex is actually the star of a single reduced marked graph of groups, which is contractible. A key tool to that end is the star graph, and the purpose of this post is to describe the star graph (for conjugacy classes) in a virtually free group and describe its usefulness.

## The reduced spine of Outer Space

Before we describe the star graph, let’s define the complex we’re working with. Given a virtually (non-abelian, to keep things interesting) free group $F$ , we’ll call the reduced spine of Outer Space $L(F)$ . In tree language, a vertex of $L(F)$ is an action of $F$ on a simplicial tree $T$ with finite stabilizers and finite quotient with the property that every edge of $T$ is surviving, see the post on shelters for a definition. Two tree actions $T$ and $T'$ are equivalent if there is an equivariant homeomorphism between them. The set of equivalence classes is partially ordered under forest collapse, and the reduced spine of Outer Space is the geometric realization of this poset.

In graphs of groups language, a vertex of $L(F)$ is a finite graph of finite groups $\mathcal{G}$ which is a union of shelters together with a marking, which we can think of as an identification of $\pi_1(\mathcal{G})$ with $F$ , well-defined up to inner automorphism, or as a homotopy equivalence in the sense of my paper $\sigma\colon \mathbb{G} \to \mathcal{G}$ from a fixed graph of finite groups $\mathbb{G}$ . Two marked graphs of groups $\tau = (\mathcal{G},\sigma)$ and $\tau' = (\mathcal{G}',\sigma')$ are equivalent if there is an isomorphism of graphs of groups $h \colon \mathcal{G} \to \mathcal{G}'$ such that $h\sigma$ is homotopic to $\sigma'$ . There is also a notion of forest collapse in graphs of groups, which partially orders the set of equivalence classes of marked graphs of groups. A marked graph of groups $\tau = (\mathcal{G},\sigma)$ is reduced if it is minimal in this partial order. The complex $L(F)$ is the geometric realization of this poset. We will adopt the graphs of groups language in this post.

## The norm

Let $\tau = (\mathcal{G},\sigma)$ be a marked graph of groups in $L(F)$ , and let $W$ be a finite set of infinite order elements of $F$ . These elements are hyperbolic in some and hence every tree action in $L(F)$ , so we may measure $\ell(w)$ , their hyperbolic translation length, and we define a norm on $\tau$ as

$\|\tau\| = \sum_{w \in W} \ell(w).$

We can represent each conjugacy class $\sigma_\sharp(w)$ in $\pi_1(\mathcal{G})$ as a cyclically reduced graph-of-groups edge path

$\gamma_w = g_0e_1g_1\ldots e_kg_k.$

We have $\ell(w) = k$ , the number of edges in $\gamma_w$ . Recall that for each vertex $v$ of $\mathcal{G}$ we have the set of directions at $v$

$D_v = \coprod_{e\in\operatorname{st}(v)} \mathcal{G}_v/\iota_e(\mathcal{G}_e) \times \{e\}.$