1Cayley graphs and the word metric

IV Topics in Geometric Group Theory

1.2 Free groups
If we want to understand a group, then a reasonable strategy now would be to
come up with a well-understood (proper geodesic) space for Γ to act (proper
discontinuously, cocompactly) on, and see what we can tell from this quasi-
isometry. There is no general algorithm for doing so, but it turns out for certain
groups, there are some “obvious” good choices to make.
Let’s begin by understanding the free group. In some sense, they are the
“simplest” groups, but they also contain some rich structure. Roughly speaking,
a free group on a set
S
is a group that is “freely” generated by
S
. This freeness
is made precise by the following universal property.
Definition
(Free group)
.
Let
S
be a (usually finite) set. A group
F
(
S
) with a
map
S F
(
S
) is called the free group on
S
if it satisfies the following universal
property: for any set map
S G
, there is a unique group homomorphism
F (S) G such that the following diagram commutes:
S F (S)
G
.
Usually, if |S| = r, we just write F (S) = F
r
.
In fancy category-theoretic language, this says the functor
F
:
Sets Grps
is left adjoint to the forgetful functor U : Grps Sets.
This definition is good for some things, but not good for others. One thing it
does well is it makes it clear that
F
(
S
) is unique up to isomorphism if it exists.
However, it is not immediately clear that
F
(
S
) exists! (unless one applies the
Thus, we must concretely construct a group satisfying this property. Apart
from telling us that
F
(
S
) actually exists, a concrete construction lets us actually
do things with the group.
For concreteness, suppose
S
=
{a
1
, . . . , a
r
}
is a finite set. We consider a
graph
X
r
=
.
.
.
where there are r “petals”. This is known as the rose with r petals.
We claim that its fundamental group is
F
(
S
). To see this, we have to
understand the universal cover
˜
X
r
of
X
r
. We know
˜
X
r
is a simply connected
graph, so it is a tree. Moreover, since it is a covering space of
X
r
, it is regular
with degree 2r on each vertex.
If r = 2, then this is our good old picture
a
1
a
1
a
2
a
2
We can use this to understand the fundamental group
π
1
(
X
r
). The elements
on
π
1
(
X
r
) are homotopy classes [
γ
] of based loops in
X
r
. Using the homotopy
lifting lemma, this map can be lifted to a path
˜γ
in the universal cover starting
at .
In general, there are many paths that represent the same element. However, there
is a “canonical” such path. Indeed, we can eliminate all backtracks in the path
to obtain an embedded path from
to the end point of
˜γ
. We may reparametrize
so that it spends equal time on each edge, but that is not important.
Algebraically, how do we understand this? Each loop in
X
r
represents an
element
a
i
in
π
1
(
X
r
). If we look at the “canonical” form of the path above, then
we see that we are writing γ in the form
γ = a
±1
i
1
a
±1
i
2
a
±1
i
3
· · · a
±1
i
n
.
The fact that we have eliminated all backtracks says this expression is reduced,
i.e. we don’t see anything looking like
a
i
a
1
i
, or
a
1
i
a
i
. Importantly, there is a
unique way to write
γ
as a reduced word, corresponding to the unique embedded
path from
to the end point of
˜γ
in the universal cover. Thus, if we somehow
expressed
γ
as a non-reduced word in the
a
1
, . . . , a
r
, then we can reduce it by
removing instances of
a
i
a
1
i
or
a
1
i
a
i
, and keep doing so until we don’t see any.
This terminates since the length of the word is finite, and the result does not
depend on the order we performed the reduction in, by uniqueness.
Example. Take the case π
1
X
2
= ha, bi. We might be given a word
a
3
a
2
b
2
b
3
a
5
a
3
a
3
b
1
.
We can reduce this to
aba
1
b
1
.
In particular, since this is not the identity, we know the original path was not
null-homotopic.
This gives us a very concrete understanding of
π
1
(
X
r
) it is generated
by
a
1
, . . . , a
r
, and each element is represented uniquely by a reduced word. To
multiply words together, write them one after another, and then reduce the
result.
From this description, it is clear that
Corollary. π
1
(
X
r
) has the universal property of
F
(
S
), with
S
=
{a
1
, . . . , a
r
}
.
So π
1
(X
r
)
=
F
r
.
Proof. Given any map f : S G, define
˜
f : F (S) G by
˜
f(a
±1
i
1
· · · a
±1
i
n
) =
˜
f(a
i
)
±1
· · ·
˜
f(a
i
n
)
±1
for any reduced word
a
±1
i
1
· · · a
±1
i
n
. This is easily seen to be well-defined and is
the unique map making the diagram commute.