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

adjoint functor theorem)

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.