Geometric Topology 20

Free Groups

The free group on one letter $a$ is the infinite cyclic group:

generated by $a$ (including $a^{-1}$). The element $a$ is not subject to any relation other than the axiom $aa^{-1} = e = a^{-1}a$, and the rule $a^ma^n= a^{n+m}$ (for any $n,m \in \mathbb{Z}$) that comes about from the meaning assigned to $a^m$ for $m>0,m=0$ and $m <0$.

Now let $G,H$ be groups, written multiplicatively. The free product $G {\Large*} H$ is a group whose elements consist of finite words made up by combining elements from either group (like $g_1h_1g_2h_2...$) in which identity elements are suppressed (or retained as a combined identity if there are no other letters). The same idea allows us to construct a free group on several letters:

Example: The free group on two letters is:

The usual product $F_1 \times F_1$ (pr $\mathbb{Z}^2 = \mathbb{Z} \oplus \mathbb{Z}$ in additive notation) is not free; its generators $a,b$ are subject to the relation $ab=ba$ or $aba^{-1}b^{-1} = e$, making it abelian.

Group Presentations

An element $aba^{-1}b^{-1}$ of a group is called the commutator of $a,b$ and is sometimes denoted $[a,b]$. We can write:

or more succinctly:

in which $a,b$ are the generators, $aba^{-1}b^{-1}=e$ is a relation and $aba^{-1}b^{-1}$ is a relator. The latter spawns other relators by taking its inverse and conjugates:

Example: The dihedral group $D_3$ of symmetries of $\triangle$ is also the symmetric group $S_3$ of permutations of $\{1,2,3\}$. It is generated (for example) by a $3$-cycle $a=(123)$ and $2$-cycle $b=(12)$, subject to the equation $bab^{-1}=a^2$ or $baba=e$. Thus:

Any group $G$ can be characterised by a number of generators (letters) $a_1,...,a_n$ and a number of relators (words) $R_1,...,R_m$ giving rise to a so-called presentation:

Normal Subgroups

The group $G = \langle a_1,...,a_n | R_1,...R_m \rangle$ is then a quotient of a free-group $F$ by a homomorphism whose kernel $N$ is the smallest normal subgroup containing the $R_i$. (Recall that a subgroup is normal if it contains all its conjugates and that $G \cong F/N$). Any element of $N$ can be simplified to the identity, using the relations.

Theorems [Nielsen-Schreier, proofs use covering spaces of graphs]:

1. Any subgroup of a free group is free, but in general the $R_i$ will not generate $N$.
2. If $F_n/N$ is finite of size $i$ then $N \cong F_{i(n-1)+1}$.

Examples:

1. For $S_3$, take $j=3$ and $R_1=a^3$, $R_2 = b^2$, $R_3 = abab$. Then $F_2/N \cong S_3$, so $N$ is a subgroup of index $i=6$ in $F_2$, and $N \cong F_7$. A graph-theoretic algorithm implies that:
   $$N = \langle a^3,b^2, abab,b^{-1}a^3b,ab^{-1}ab,a^{-1}ba^{-1}b,b^{-1}aba \rangle.$$
2. The kernel of the homomorphism $F_2 \rightarrow F_1 \times F_1$ is the normal subgroup generated by all the commutators $xyx^{-1}y^{-1}$ with $x,y \in F_2$, and $F_1 \times F_1 \cong F_2 / [F_2,F_2]$ is the abelisanisation of $F_2$.

Simplifying Relations

Let $K$ be a (left or right) trefoil knot in $\mathbb{R}^3$. The so-called Wirtinger presentation of $G = \pi_1(\mathbb{R}^3 - K)$ is given by:

with one relation for each crossing. This is explained with some previous work, however we are only concerned with the group theory here.

In $G$ we have $\alpha^{-1}\beta \alpha = \gamma = \beta \alpha \beta^{-1}$, which implies the third equation $\gamma^{-1}\alpha \gamma = \beta$. So we can dispense with $\gamma$, and observe that:

Set $a = \alpha \beta$ and $b = \alpha \beta \alpha$. Then $a^3 = b^2$. Since $\alpha = a^{-1}b$ and $\beta = b^{-1}a^2$,

This presentation reflects the realisation of a trefoil knot wrapping around a torus.