# The Star Graph

April 02, 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

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

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 $v$*

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.

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.

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

## 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.