The Star Graph

02 Apr 2022

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 \(v\)

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

There is an obvious left action of \(\mathcal{G}_v\) on \(D_v\) which descends to an action on the set of pairs of directions at \(v\). A turn is an orbit under this action. A turn is degenerate if every pair of directions representing it are identical and is nondegenerate otherwise. The vertex group elements of the path \(\gamma_w\) are not quite well-defined if the edge groups of \(\mathcal{G}\) are nontrivial, but the turns \([\{(\iota_{\bar e_i}(\mathcal{G}_{\bar e_i}),\bar e_i), (g_i\iota_{e_{i+1}}(\mathcal{G}_{e_{i+1}}),e_{i+1})\}]\) for \(1 \le i \le k-1\) and \([\{(g_k^{-1}\iota_{\bar e_k}(\mathcal{G}_{e_k}),e_k), (g_0\iota_{e_1}(\mathcal{G}_{e_1}),e_1)\}]\) are well-defined. We say that \(\gamma_w\) crosses these turns, which are nondegenerate because \(\gamma_w\) is cyclically reduced.

The star graph

We define the star graph of a marked graph of groups \(\tau = (\mathcal{G},\sigma)\) with respect to \(W\). Its vertex set is the union of the sets \(D_v\) as \(v\) varies over the vertices of \(\mathcal{G}\). For each turn crossed by some \(\gamma_w\), the star graph contains \(|\mathcal{G}_v|\) edges, each one connecting a pair of directions representing this turn. Since the turns crossed by \(\gamma_w\) are nondegenerate, the star graph contains no loop edges.

A star graph

In the above example we have

\[F = C_4*_{C_2} C_4 * C_3 = \langle s, t, u \mid s^4 = t^4 = u^3 = 1,\ s^2 = t^2 \rangle,\]

and the marked graph of groups \(\tau = (\mathcal{G},\sigma)\) is the “obvious” graph of groups presentation of \(F\). There are three vertices with vertex groups \(\langle s\rangle\), \(\langle t\rangle\) and \(\langle u\rangle\). There are two edges \(a\) and \(b\). The vertex \(v\) with vertex group \(\langle t\rangle\) has valence two; both \(a\) and \(b\) are incident to it in the positive orientation. The edge \(a\) connects \(v\) to the vertex with vertex group \(\langle s\rangle\) and has \(C_2\) edge group, while \(b\) has trivial edge group. The set \(W\) is the singleton \(\{stu^2t^2\}\). The star graph of of \(\tau\) with respect to \(W\) has eleven vertices. The component corresponding to \(\langle s\rangle\) has two vertices, and there are four edges corresponding to the turns in the orbit \([\{(\langle s^2\rangle,\bar a),(s\langle s^2\rangle,\bar a)\}]\). The component corresponding to \(\langle u\rangle\) has three vertices, and there are three edges corresponding to the turns in the orbit \([\{(1,\bar b),(u^2,\bar b)\}]\). The component corresponding to the vertex \(v\) has six vertices, two for directions with underlying oriented edge \(a\) and two with underlying oriented edge \(b\). This subgraph of the star graph is complete bipartite corresponding to the turns in the orbits \([\{(\langle t^2\rangle,a),(t,b)\}]\) and \([\{(1,b),(\langle t^2\rangle,a)\}]\).

We can compute the norm of \(\tau\) from the star graph by summing up

\[\|\tau\| = \frac{1}{2}\sum_{v \in \mathcal{G}}\sum_{d \in D_v} \frac{\operatorname{valence}(d)}{|\mathcal{G}_v|}.\]

So in this example we see that we have \(\|\tau\| = 4\).

Dot product and absolute value

Following Culler and Vogtmann, given subsets \(S\) and \(T\) of the vertex set of a graph, define the dot product \(S\cdot T\) to be the number of (unoriented) edges in the graph with one vertex in \(S\) and the other in \(T\). The absolute value \(|S|\) of \(S\) is the dot product of \(S\) with its complement. An ideal edge \(\alpha\) is in particular a subset of the vertex set of the star graph, so we may compute \(|\alpha|\). If \(\alpha' = g.\alpha\) under the action of \(\mathcal{G}_v\) on \(D_v\), then \(|\alpha| = |\alpha'|\). Similarly a direction \(d \in D_v\) is a singleton subset of the vertex set of the star graph and we may compute \(|d| = |\{d\}|\). If \(d' = g.d\), then \(|d| = |d'|\). Put another way, the absolute value of \(d\) is a property of the underlying oriented edge \(e\) and we may sometimes write \(|e|\) for \(|d|\).

In the above example, the ideal edge \(\alpha = \{(\langle t^2\rangle,a),(t,b),(t^3,b)\}\) has absolute value \(|\alpha| = 4\), while the direction \(d = (\langle t^2\rangle,a)\) also has absolute value \(4\). Both \(d\) and \(\alpha\) have edge group isomorphic to \(C_2\), so we have

\[\frac{|d|}{|\mathcal{G}_a|} = \frac{|\alpha|}{|\mathcal{G}_\alpha|} = 2.\]

If we blow up the ideal edge \(\alpha\) obtaining \(\tau^\alpha = (\mathcal{G}^\alpha,\sigma^\alpha)\), the original \(\gamma_w\) was \(as\bar atb u^2\bar b t^2\) and the new \(\gamma_w\) is \(as\bar ab tu^2t^{-1}\bar b \bar\alpha t\alpha t^2\), which has \(2\) more edges. If we then collapse the collapsible edge \(a\) corresponding to the direction \(d \in D(\alpha)\) to obtain \(\tau^\alpha_a\), the new \(\gamma_w\) is \(sbtu^2t^{-1}\bar b\bar\alpha t\alpha s^2\), which has \(2\) fewer edges.

The marked graphs of groups obtained by blowing up and then collapsing

This illustrates a general principal:

Proposition (Krstić–Vogtmann). If \(\Phi = \{\alpha_1,\ldots,\alpha_k\}\) is a set of pairwise compatible ideal edges in a reduced marked graph of groups \(\tau = (\mathcal{G},\sigma)\) and \(\{e_1,\ldots,e_k\}\) are a set of oriented edges in \(\mathcal{G}\) which form a forest in \(\tau^{\alpha_1,\ldots,\alpha_k}\), we have

\[\|\tau^{\alpha_1,\ldots,\alpha_k}_{e_1,\ldots,e_k}\| = \|\tau\| + \sum_{i=1}^k \frac{|\alpha_i|}{|\mathcal{G}_{\alpha_i}|} - \sum_{i=1}^k \frac{|e_i|}{|\mathcal{G}_{e_i}|}.\]

Wrapping up

The star of a reduced marked graph of groups \(\tau\) is the set of marked graphs of groups \(\tau'\) which collapse onto \(\tau\). The ball of radius \(r\) is the union of the stars of reduced marked graphs of groups \(\tau\) with \(\|\tau\| \le r\). What Krstić–Vogtmann show is that as long as the ball of radius \(r' < r\) is nonempty, the ball of radius \(r\) deformation retracts onto the ball of radius \(r'\). They show that for well-chosen \(W\), there is a single reduced marked graph of groups of minimal norm, thus proving that \(L\) is contractible. To do both of these steps, they make extensive use of the star graph to analyze the effect of blowing up and collapsing edges in marked graphs of groups.