# Ideal Edges

March 27, 2022

Let’s once again fix a group $G$
that acts cocompactly on a tree $T$
.
In the previous post we defined what it meant for an edge of $T$
to be *surviving* or *collapsible.*
Collapsing orbits of collapsible edges in $T$
yields
a new tree $T'$
which we say is obtained from $T$
by forest collapse.
Two trees $T$
and $T'$
are in the same *deformation space*
if there are $G$
-equivariant maps $T \to T'$
and $T' \to T$
.
The set of $G$
-equivariant homeomorphism classes of trees in the same deformation space as $T$
all of whose edges are surviving
is partially ordered under the operation of forest collapse,
and this partial order contains minimal elements, which are *reduced* trees.
To study this poset, or its geometric realization $L_T(G)$
,
it would be convenient to understand all trees collapsing onto a given reduced tree.
This is captured in the idea of an *ideal edge,*
which I’d like to describe in this post.
The idea of an ideal edge goes back to Culler and Vogtmann’s
original paper on Outer Space;
the treatment here is adapted from a paper of Krstić and Vogtmann.

Again we will work in the quotient graph of groups $\mathcal{G}$
,
the construction of which we described in the previous post.
We assume that $\mathcal{G}$
is finite and connected,
and that if a vertex $v$
has valence less than 3,
then at least one of the edge-to-vertex group inclusions
$\iota_e\colon\mathcal{G}_e \to \mathcal{G}_v$
is not surjective.
In tree language this says that $T$
is minimal and has no subdivision vertices.
The graph of groups $\mathcal{G}$
is *marked.*
If we fix a tree $T$
in the deformation space
and identify $G$
with the fundamental group of its quotient graph of groups $\mathbb{G}$
,
we may think of the marking as a *homotopy equivalence* in the sense of my paper
$\sigma\colon \mathbb{G} \to \mathcal{G}$
.

## Directions

Let $v$
be a vertex of $\mathcal{G}$
.
Let $\operatorname{st}(v)$
be the set of oriented edges of $\mathcal{G}$
with initial vertex $v$
.
The set of *directions* at $v$
, denoted $D_v$
, is the set

The group $\mathcal{G}_v$ acts on $D_v$ on the left, and under the identification of $\mathcal{G}$ with the quotient graph of groups of $T$ , if $\tilde v$ is a lift of $v$ to $T$ , there is a $\mathcal{G}_v$ -equivariant bijection between $D_v$ and the set $\operatorname{st}(\tilde v)$ .

Suppose $T'$ is a tree such that collapsing the orbit of an edge $\tilde e$ incident to $v$ yields a tree equivariantly homeomorphic to $T$ . If we suppose that $T'$ is minimal, has no subdivision vertices and that every edge of $T'$ is surviving, we can make the following observations:

- $\tilde e$ separates $\operatorname{st}(\tilde v)$ into two components, each of which has at least two edges, since $T'$ is minimal and has no subdivision vertices.
- The stabilizer of $\tilde e$ is a subgroup of the stabilizer of $\tilde v$ and is equal to the stabilizer of the new vertex $\tilde w$ . If an edge in $\operatorname{st}(\tilde v)$ now has initial vertex $\tilde w$ , so does every edge in its $\operatorname{stab}(\tilde e)$ -orbit and no others.
- Since $\tilde e$ is surviving, another edge incident to $\tilde w$ must be collapsible; in other words $\operatorname{stab}(\tilde e) = \operatorname{stab}(\tilde e')$ for some edge $\tilde e'$ with initial vertex $\tilde w$ . For the edge $\tilde e'$ to be collapsible, we need that its terminal vertex is not in the orbit of $\tilde w$ .

Furthermore conversely, if we have a subset of $\operatorname{st}(\tilde v)$ satisfying the above three properties, we can “blow up” the tree $T$ , inserting a new edge orbit $\tilde e$ .

## Ideal edges

The idea of an ideal edge in $\mathcal{G}$
is designed to capture the above observations.
To wit, an *ideal edge* based at a vertex $v$
is a subset $\alpha \subset D_v$
that is required to satisfy the following conditions.
Let $\mathcal{G}_\alpha$
be the subgroup of $\mathcal{G}_v$
generated by the subgroups $g\iota_e(\mathcal{G}_e)g^{-1}$
for each direction $(g\iota_e(\mathcal{G}_e),e)$
contained in $\alpha$
.

- There are at least two directions in $\alpha$ and at least two directions not in $\alpha$ .
- If $(g\iota_e(\mathcal{G}_e),e) \in \alpha$ , then the intersection $\mathcal{G}_v(g\iota_e(\mathcal{G}_e),e) \cap \alpha$ equals $\mathcal{G}_\alpha(g\iota_e(\mathcal{G}_e),e)$ .
- The subgroup $\mathcal{G}_\alpha$ is equal to $g\iota_e(\mathcal{G}_e)g^{-1}$ for some direction $(g\iota_e(\mathcal{G}_e),e) \in \alpha$ and no direction with underlying oriented edge $\bar e$ is in $\alpha$ .

We will say an oriented edge $e \in \alpha$
if there is some direction
in $\alpha$
with underlying oriented edge $e$
.
Notice that it follows from the conditions that $\alpha$
contains at least two oriented edges.
Let $D(\alpha)$
be the set of oriented edges supporting a direction
satisfying the third item above.
If $\mathcal{G}$
is reduced and $\alpha$
is an ideal edge
satisfying $\mathcal{G}_\alpha = \mathcal{G}_v$
,
then $D_v - \alpha$
is also an ideal edge.
The first item in the definition is clear.
The second says that for each oriented edge $e \in \operatorname{st}(v)$
,
either all directions with underlying oriented edge $e$
are in $\alpha$
or none are,
which is also true of $D_v - \alpha$
.
Since $\mathcal{G}$
is reduced, some and hence every edge $e \in D(\alpha)$
must form a loop based at $v$
(for otherwise we could collapse $e$
),
so we have $\bar e \in D(D_v - \alpha)$
.
We call $D_v - \alpha$
the *inverse* of $\alpha$
.
Two ideal edges $\alpha$
and $\alpha'$
are *equivalent*
if there exists $g \in \mathcal{G}_v$
such that $g.\alpha = \alpha'$
.

An ideal edge $\alpha$
based at $v$
is *contained in* an ideal edge $\beta$
if there exists $g \in \mathcal{G}_v$
such that $g.\alpha \subset \beta$
.
If $\alpha$
and $\beta$
are based at $v$
,
the ideal edges are *disjoint* if $g.\alpha \cap \beta = \varnothing$
for all $g \in \mathcal{G}_v$
.
If $\alpha$
and $\beta$
are based at different vertices, then they are disjoint.
We say $\alpha$
and $\beta$
are *compatible*
if one is properly contained in the other or if they are disjoint and not inverse.

## Blowing up ideal edges

Let $\alpha$
be an ideal edge of the marked graph of groups $\tau = (\mathcal{G},\sigma)$
.
We construct a new marked graph of groups $\tau^\alpha = (\mathcal{G}^\alpha,\sigma^\alpha)$
by *blowing up* $\alpha$
as follows.
The vertices and edges of $\mathcal{G}^\alpha$
are the same as the vertices and edges of $\mathcal{G}$
with one new vertex $v_\alpha$
and one new edge $\alpha$
;
the other vertices and edges are *old.*
The old vertex and edge groups are equal to what they are in $\mathcal{G}$
.
We have $\mathcal{G}^\alpha_{v_\alpha} = \mathcal{G}_\alpha = \mathcal{G}^\alpha_\alpha$
.
The new edge $\alpha$
begins at $v_\alpha$
and ends at $v$
.
If $e$
is an old oriented edge in $\alpha$
,
then $e$
now has initial vertex $v_\alpha$
and the inclusion $\iota_e\colon \mathcal{G}^\alpha_e \to \mathcal{G}^\alpha_\alpha$
is obtained from $\iota_e\colon \mathcal{G}_e \to \mathcal{G}_v$
by recalling that we have $g\iota_e(\mathcal{G_e})g^{-1} \le \mathcal{G}_\alpha$
for some $g \in \mathcal{G}_v$
.
Note that there is a choice here of group element $g$
;
differing choices will yield *equivalent* markings in a sense that I won’t make precise here.
All other oriented edges have their initial vertices and edge-to-vertex group inclusions unchanged.
There is a *collapse map* $\mathcal{G}^\alpha \to \mathcal{G}$
defined by collapsing the edge $\alpha$
and then “twisting” the edges incident to $v_\alpha$
by the corresponding choice of group element $g \in \mathcal{G}_v$
.
We’ll describe maps of general graphs of groups in another post.
Choose a homotopy inverse $f\colon \mathcal{G} \to \mathcal{G}^\alpha$
and define $\sigma^\alpha = f\sigma$
.

Observe that if $\beta$ is disjoint from (and not inverse to) or properly included in $\alpha$ , then $\beta$ may be regarded as an ideal edge of $\mathcal{G}^\alpha$ . If $\alpha$ are disjoint and not inverse, it’s not hard to see that $(\tau^\alpha)^\beta = (\tau^\beta)^\alpha$ .

An *oriented ideal forest* in a marked graph of groups $\tau$
is a collection of equivalence classes $\{\alpha_1,\ldots,\alpha_k\}$
of ideal edges in $\tau$
that are pairwise compatible.
The ideal edges in an oriented ideal forest are partially ordered under inclusion.
By repeatedly blowing up maximal elements of this partial order,
we obtain a marked graph of groups $\tau^{\alpha_1,\ldots,\alpha_k}$
.
Because each ideal edge $\alpha$
in a reduced marked graph of groups $\tau$
has an oriented edge in $D(\alpha)$
,
it follows that each edge of $\tau^{\alpha_1,\ldots,\alpha_k}$
is surviving.

A pair of ideal edges $\alpha$
and $\beta$
are *pre-compatible*
if they are compatible, or $\alpha$
is invertible with inverse $\bar\alpha$
and $\bar\alpha$
is included in $\beta$
.
It follows that $\beta$
is invertible and that $\bar\beta$
is included in $\alpha$
.
An *ideal forest* is a collection of pairwise pre-compatible ideal edges
containing the inverse of each of its invertible elements.
We can complete each oriented ideal forest $\Phi$
to an ideal forest $\Phi^\pm$
by adding
the inverses of each of its invertible elements.
Krstić and Vogtmann prove for graphs of finite groups
as Lemma 5.7 of their paper
that if $\Phi^\pm = \Psi^\pm$
as ideal forests
in a reduced marked graph of groups $\tau$
,
we have $\tau^\Phi = \tau^\Psi$
as marked graphs of groups.
Again for graphs of finite groups,
Krstić and Vogtmann prove as Proposition 5.9 that the poset of marked graphs of groups in $L_T(G)$
collapsing to a reduced marked graph of groups $\tau$
is isomorphic to the poset of ideal forests in $\tau$
.