Free groups - basics
B.Sury
Stat-Math Unit
Indian Statistical Institute
Bangalore, India
AIS - May 2010 - IIT Bombay
“Only that thing is free which exists by the necessities of its own nature,
and is determined in its actions by itself alone” - Baruch Spinoza
“Everything that is really great and inspiring is created by the individual
who can labor in freedom” - Albert Einstein
“Now go we in content
To liberty, and not to banishment.”
... W.Shakespeare (As You Like It)
“Freedom is like drink. If you take any at all, you might as well take
enough to make you happy for a while” - Finley Peter Dunne
1

Introduction
Free objects in a category (whatever these animals may be) are the most basic
objects in mathematics.
A paradigm is the theory of free groups.
They arose
naturally while studying the geometry of hyperbolic groups but their fundamental
role in group theory was recognized by Nielsen (who named them so), Dehn and
others. In these lectures, we introduce free groups and their subgroups and study
their basic properties.
The lectures by Professor Anandavardhanan would treat
the more general notions of free products with amalgamation and HNN extensions
in detail. However, it is beneficial to look at them already for our purposes as they
bear repetition. We shall do so briefly in the course of these lectures.
The notions of free groups, free products, and of free products with amalga-
mation come naturally from topology. For instance, the fundamental group of
the union of two path-connected topological spaces joined at a single point
is isomorphic to the so-called free product of the individual fundamental
groups.
(More generally) The Seifert-van Kampen theorem asserts that if
X
=
V
∪
W
is a union of path-connected spaces with
V
∩
W
non-empty and path-
connected, and if the homomorphisms
π
1
(
V
∩
W
)
→
π
1
(
V
) and
π
1
(
V
∩
W
)
→
π
1
(
W
) induced by inclusions, are injective, then
π
1
(
X
) is isomorphic to the
so-called free product of
π
1
(
V
) and
π
1
(
W
) amalgamated along
π
1
(
V
∩
W
).
Many naturally occurring groups can be viewed in terms of these construc-
tions.
For instance,
SL
(2
,
Z
)
is the free product of
Z
/
4
and
Z
/
6
amalgamated along
a subgroup isomorphic to
Z
/
2
.
The fundamental group of the Klein bottle is isomorphic to the free product
of two copies of
Z
amalgamated along
2
Z
.
The so-called HNN extensions also have topological interpretations. Suppose
V
and
W
are open, path-connected subspaces of a path-connected space
X
and suppose that there is a homeomorphism between
V
and
W
inducing
isomorphic embeddings of
π
1
(
V
) and
π
1
(
W
) in
π
1
(
X
).
One constructs a
space
Y
by attaching the handle
V
×
[0
,
1] to
X
, identifying
V
× {
0
}
with
V
and
V
× {
1
}
with
W
. Then, the fundamental group
π
1
(
Y
) of
Y
is the HNN
extension of
π
1
(
V
) relative to the isomorphism between its subgroups
π
1
(
V
)
and
π
1
(
W
).