# From Free Splittings to Graphs of Groups

January 06, 2022

In the previous post, we defined *free splittings* for a group $F$
.
The quotient of a free splitting $T$
of $F$
by the action of $F$
is a *graph of groups with trivial edge groups.*
The purpose of this post is to introduce *maps* of graphs of groups with trivial edge groups
and explore lifting maps of the quotient graph of groups to twisted equivariant maps
of the tree $T$
.

## Graphs of groups with trivial edge groups

Let $G$
be a finite graph.
The structure of a *graph of groups with trivial edge groups* on $G$
is the assignment, for each vertex $v$
of $G$
, of a group $\mathcal{G}_v$
.
We write $\mathcal{G}$
for the graph of groups.
Associated to $\mathcal{G}$
we can form a *graph of spaces* $X_{\mathcal{G}}$
à la Scott and Wall.
For each vertex $v$
of $G$
, take a CW complex $X_v$
with one vertex $\star_v$
that is a $K(\mathcal{G}_v,1)$
.
The space $X_{\mathcal{G}}$
is formed from the disjoint union of these CW complexes
by adding, for each edge $e$
of $G$
with initial vertex $v$
and terminal vertex $w$
,
an edge connecting $\star_v$
to $\star_w$
.
There is a natural “retraction” $r\colon X_{\mathcal{G}} \to G$
that collapses each $K(\mathcal{G}_v,1)$
to the vertex $\star_v$
.

The *fundamental group of the graph of groups $\mathcal{G}$*
is the fundamental group of the graph of spaces $X_{\mathcal{G}}$
.
If we choose a basepoint $p \in X_{\mathcal{G}}$
in the image of the retraction,
we may homotope each element of $\pi_1(X_{\mathcal{G}},p)$
into the $1$
-skeleton,
and further to an *edge path in $\mathcal{G}$* of the form

where $e'_1$
and $e'_{k+1}$
are terminal and initial segments
of oriented edges $e_1$
and $e_{k+1}$
of $G$
,
respectively, where $e_2,\ldots,e_k$
are oriented edges of $G$
,
where $g_i$
for $1 \le i \le k$
is an element of $\mathcal{G}_{v_i} = \pi_1(X_{v_i},\star_{v_i})$
,
and where $v_i$
is the terminal vertex of the edge $e_i$
and the initial vertex of the edge $e_{i+1}$
.
Notice that this notion makes sense without reference to $X_{\mathcal{G}}$
.
Homotopy rel endpoints for edge paths in $\mathcal{G}$
has a combinatorial flavor
that makes no reference to $X_{\mathcal{G}}$
:
it is generated by performing multiplication or factoring in vertex groups
and adding or removing subpaths of the form $e \bar e$
, where $\bar e$
is the edge
$e$
in its opposite orientation.
An edge path is *tight* if the number of edges in the path cannot be decreased by a homotopy.
Every edge path is homotopic to a unique tight path (which may contain no edges).

Bass–Serre theory asserts that $F = \pi_1(\mathcal{G},p) = \pi_1(X_{\mathcal{G}},p)$ is a free product of the form

$F = A_1 * \cdots * A_n * F_k,$Where the $A_i$ are the nontrivial vertex groups of $\mathcal{G}$ and $F_k$ , a free group of rank $k$ , is the ordinary fundamental group of $G$ .

## The Bass–Serre tree

Given a graph of groups with trivial edge groups $\mathcal{G}$
and a point $p \in G$
,
we wish to define a free splitting $\Gamma$
of $F$
equipped with a basepoint $\tilde p$
called the *Bass–Serre tree* of $\mathcal{G}$
.
Given a vertex $v$
of $G$
, write $[p,v]$
for the set of homotopy classes
of edge paths in $\mathcal{G}$
beginning at $p$
and ending at $v$
.
There is a natural right action of $\mathcal{G}_v$
on $[p,v]$
defined by $\gamma.g = \gamma g$
.
Let $V$
be the set of vertices of $G$
.
The vertex set of $\Gamma$
is the set

Two vertices $[\gamma] \mathcal{G}_v$ and $[\gamma']\mathcal{G}_w$ are adjacent in $\Gamma$ if and only if the path $\bar\gamma\gamma'$ is homotopic to a path of the form $g e g'$ , i.e. a path of length one. One can check that this is well-defined independent of the choice of representatives $\gamma$ and $\gamma'$ . There is a natural action of $F = \pi_1(\mathcal{G},p)$ on $\Gamma$ defined as $[\gamma].[\sigma]\mathcal{G}_v = [\gamma\sigma]\mathcal{G}_v$ . The fundamental theorem of Bass–Serre theory asserts that $\Gamma$ is a tree, that the $F$ -action has trivial edge stabilizers, and that the “quotient graph of groups” $F\backslash\backslash\Gamma$ , which I won’t describe here, is naturally identified with $\mathcal{G}$ . Let $\pi\colon \Gamma \to G$ be the natural projection.

Suppose $\gamma$ is a tight path in $\mathcal{G}$ beginning at a vertex $v$ and ending at a vertex $w$ . A lift $\tilde v$ of $v$ to $\Gamma$ is determined by a tight path $\sigma$ from $p$ to $v$ . The path $\gamma$ lifts uniquely to a tight path $\tilde\gamma$ in $\Gamma$ beginning at $\tilde v$ : it is the unique tight path connecting the vertex $[\sigma]\mathcal{G}_v$ to the vertex $[\sigma\gamma]\mathcal{G}_w$ . More generally it is possible to lift edge paths that are not tight or do not begin and end at vertices. In particular lifting the trivial path at $p$ defines a basepoint $\tilde p$ in $\Gamma$ .

## Maps of graphs of groups with trivial edge groups

Let $\mathcal{G}$
be a graph of groups with trivial edge groups
and $X_{\mathcal{G}}$
an associated graph of spaces.
A *map of graphs of groups with trivial edge groups*
$f\colon \mathcal{G} \to \mathcal{G}$
is a pair of maps $f_X\colon X_{\mathcal{G}} \to X_{\mathcal{G}}$
and $f\colon G \to G$
such that the following diagram commutes

$$\require{AMScd}\begin{CD} X_{\mathcal{G}} @>f_X» X_{\mathcal{G}} \ @VrVV @VVrV \ G @>f» G. \end{CD}$$

A *homotopy* of maps of graphs of groups with trivial edge groups
is a pair of homotopies such that the relevant diagram commutes.
A map $f\colon \mathcal{G} \to \mathcal{G}$
is a *homotopy equivalence*
if, as usual, there is a map $g\colon \mathcal{G} \to \mathcal{G}$
such that $gf$
and $fg$
are each homotopic to the identity map.

It is not too hard to see that each map $f\colon \mathcal{G} \to \mathcal{G}$ is homotopic to one that sends the vertices $\star_v$ of $X_{\mathcal{G}}$ to vertices of $X_{\mathcal{G}}$ and further sends edges of $G \subset X_{\mathcal{G}}$ to (possibly trivial) edge paths in $\mathcal{G}$ .

Like edge paths, a map of graphs of groups with trivial edge groups that is a homotopy equivalence has a combinatorial shadow in $\mathcal{G}$ that does not depend on $X_{\mathcal{G}}$ : the data of $f$ is a continuous map $f\colon G \to G$ taking vertices to vertices and edges to edge paths, along with for each edge $e$ of $G$ , an edge path $f(e)$ in $\mathcal{G}$ , and for each vertex $v$ of $G$ with nontrivial vertex group, an isomorphism $f_v\colon \mathcal{G}_v \to \mathcal{G}_{f(v)}$ . Homotopy for maps of graphs of groups has a combinatorial shadow in $\mathcal{G}$ as well; I won’t describe it.

Suppose $f\colon \mathcal{G} \to \mathcal{G}$ is a homotopy equivalence of the form above. Given a basepoint $p \in G$ and a path $\sigma$ from $p$ to $f(p)$ , the homotopy equivalence $f$ induces an automorphism $\Phi \colon \pi_1(\mathcal{G},p) \to \pi_1(\mathcal{G},p)$ by the rule

$\Phi([\gamma]) = [\sigma f(\gamma) \bar\sigma].$If $\Gamma$
is the Bass–Serre tree of $\mathcal{G}$
with basepoint $\tilde p$
lifting $p$
,
the choice of path $\sigma$
also defines a lift
$\tilde f\colon \Gamma \to \Gamma$
defined as follows:
A point $\tilde x$
of $\Gamma$
is determined by an edge path $\gamma$
in $\mathcal{G}$
beginning at $p$
:
if $\tilde x$
is a vertex, the path $\gamma$
is any path in the $\mathcal{G}_x$
-orbit
$[\gamma]\mathcal{G}_x$
determining $\tilde x$
.
In this case, the point $\tilde f(\tilde x)$
is $[\sigma f(\gamma)]\mathcal{G}_{f(x)}$
.
One checks that this is independent of the choice of $\gamma$
.
This can be extended to define $\tilde f$
for points that are not vertices.
The map $\tilde f$
is $\Phi$
-*twisted equivariant* in the sense that

for each point $\tilde x \in \Gamma$
and element $g$
of $\pi_1(\mathcal{G},p)$
.
We say that $\tilde f$
*corresponds to $\Phi$
.*

## A little about the map $D\tilde f$

Continuing notation from the previous paragraph,
let $\tilde v$
be the vertex $[\gamma]\mathcal{G}_v$
of $\Gamma$
.
Suppose the final vertex group element of $\gamma$
is trivial.
A lift $\tilde f\colon \Gamma \to \Gamma$
of $f$
determined by the choice of path $\sigma$
from $p$
to $f(p)$
satisfies $\tilde f(\tilde v) = \tilde v$
if and only if
$f$
fixes $v$
and
$\sigma$
is homotopic to a path of the form $\gamma h f(\bar\gamma)$
for $h \in \mathcal{G}_v$
.
A *direction at $\tilde v$* is an edge $\tilde e$
with initial vertex $\tilde v$
.
The map $g \mapsto [\gamma g\bar\gamma]$
determines an isomorphism from $\mathcal{G}_v$
to the stabilizer of $\tilde v$
in $\pi_1(\mathcal{G},p)$
.
Under this isomorphism there is a (left) action of $\mathcal{G}_v$
on the set of directions based at $\tilde v$
.
Let $\operatorname{st}(v)$
be the set of edges of $G$
with initial vertex $v$
.
There is a $\mathcal{G}_v$
-equivariant bijection from the set of directions based at $\tilde v$
with the set

defined as follows. Let $\tilde w$ be the terminal vertex of the edge $\tilde e$ . The map sends $\tilde e$ to the pair $(g,e)$ if we have $\tilde w = [\gamma ge]\mathcal{G}_w$ . The map $\tilde f\colon \Gamma \to \Gamma$ induces a map $D\tilde f$ of the set of directions at $\tilde v$ as follows: $D\tilde f(\tilde e)$ is the first edge on the edge path $\tilde f(\tilde e)$ . We have

$$\tilde f(\tilde w) = [\sigma f(\gamma ge)]\mathcal{G}*{f(w)}
= [\gamma h f(\bar\gamma)f(\gamma)f_v(g)f(e)]\mathcal{G}*{f(w)}
= [\gamma h f_v(g) f(e)]\mathcal{G}{f(w)}.$$

If the edge path $f(e)$ begins $g_0 e_1$ , we see that $D\tilde f(g,e) = (hf_v(g)g_0,e_1)$ .

## My latest point of confusion

Suppose $\tilde v$ has infinite valence, or equivalently that the group $\mathcal{G}_v$ is infinite. Let’s assume $\mathcal{G}_v$ is countable. To get the result that I want, I’m happy to restrict further, and I’m particularly interested in the case where $\mathcal{G}_v$ is virtually finite-rank free.

Suppose $\tilde f$
is a lift of $f\colon \mathcal{G} \to \mathcal{G}$
that fixes $\tilde v$
.
We say that $\tilde f$
is *principal* if $\operatorname{Fix}(\tilde f)$
contains at least two points
or if $D\tilde f$
fixes a direction based at $\tilde v$
.
Two lifts $\tilde f$
and $\tilde f'$
fixing $\tilde v$
corresponding to elements $h$
and $h'$
in $\mathcal{G}_v$
are *isogredient*
if there exists $c \in \mathcal{G}_v$
such that
the maps $D\tilde f$
and $D\tilde f'$
satisfy $D\tilde f' = c D\tilde f c^{-1}$
:
that is,
for a direction $(x,e)$
,
we have $(h' f_v(x) g_0,e_1) = (ch f_v(c^{-1}) f_v(x) g_0,e_1)$
,
which holds if and only if $ h' = ch f_v(c^{-1})$
.
It’s not too hard to see that if $(x,e)$
is a fixed direction for $D \tilde f$
,
then $cx$
is a fixed direction for $D\tilde f'$
,
or more generally if $(x,e)$
is a periodic direction for $D\tilde f$
,
then $cx$
is a periodic direction for $D\tilde f'$
.
We are interested in principal lifts up to isogredience,
so we may simplify things by noting that up to isogredience, if there is a fixed point,
we may assume that it is of the form $(1,e)$
.
In this case, we have

so we conclude that $h = g_0^{-1}$ . Therefore up to isogredience, fixed directions are pretty simple. Periodic directions are a little more complicated: assuming that $e_1 = e$ , the condition that $(1,e)$ is mapped to itself by $D\tilde f^k$ is that

$h f_v(h)f_v^2(h)\ldots f_v^{k-1}(h)f_v^{k-1}(g_0)f_v^{k-2}(g_0) \ldots f_v(g_0)g_0 = 1.$What I want is sufficient conditions on $\mathcal{G}_v$ to guarantee the existence of $k > 0$ independent of the choice of lift $\tilde f$ such that if $D\tilde f^k$ has a periodic direction, then that direction is fixed. As already noted, we may simplify up to isogredience by assuming that the identity is periodic.