# Fundamental groups of 3-dimensional spherical space forms

## 1 Introduction


## 2 Finite subgroups of SO(4)

To determine finite subgroups of $SO(4)$$SO(4)$ it is necessary to proceed in three steps:

• determine finite subgroups of $SO(3)$$SO(3)$,
• use the covering map $S^3 \to SO(3)$$S^3 \to SO(3)$ to determine finite subgroups of $S^3$$S^3$,
• use the fact that $SO(4)$$SO(4)$ is doubly covered by $S^3 \times S^3$$S^3 \times S^3$ to determine its finite groups.

### 2.1 Finite subgroups of SO(3)

To classify finite subgroups of $SO(3)$$SO(3)$ one has to analyse action of these groups on $S^2$$S^2$. From Riemann-Hurwitz formula we obtain the following equation

$\displaystyle 2 \left(1 - \frac{1}{N} \right) = \sum_{i = 1}^q \left(1 - \frac{1}{n_i} \right),$

where $N$$N$ denotes the order of the group, $q$$q$ denotes number of orbits with non-trivial isotropy and $n_i$$n_i$ denotes the order of the respective isotropy subgroup. Solutions to this equation yield the desired list.

Theorem 2.1 [Wolf2011, thm 2.6.5.]. Every finite subgroup of $SO(3)$$SO(3)$ is either

These groups are called polyhedral groups.

### 2.2 Finite subgroups of S^3

Let $\mathbb{H}$$\mathbb{H}$ denote the algebra of quaternions and treat $S^3$$S^3$ as a subset of $\mathbb{H}$$\mathbb{H}$ of quaternions of norm $1$$1$. Consider an action of $S^3$$S^3$ on $\mathbb{H}$$\mathbb{H}$ by conjugation

$\displaystyle q \mapsto (q' \mapsto q \cdot q' \cdot q^{-1}).$

This action preserves $1$$1$, so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation $\pi \colon S^3 \to SO(3)$$\pi \colon S^3 \to SO(3)$ with kernel equal to $\{\pm 1\}$$\{\pm 1\}$.

If $G$$G$ is a finite subgroup of $S^3$$S^3$, then let $F = \pi(G)$$F = \pi(G)$. If $F = G$$F = G$, then, since $-1$$-1$ is the only element of $S^3$$S^3$ of order $2$$2$, $F$$F$ and $G$$G$ must be both of odd order. Therefore comparing this with the list of finite subgroups of $SO(3)$$SO(3)$ yields that $F$$F$ and $G$$G$ are both cyclic of odd order. On the other hand, if $F \neq G$$F \neq G$, then $G$$G$ is an extension of the form

$\displaystyle 1 \to \{\pm 1\} \to G \to F \to 1.$

These considerations yields the following theorem.

Theorem 2.2 [Wolf2011]. Every finite subgroup of $S^3$$S^3$ is either

These groups are called binary polyhedral groups.

### 2.3 Finite subgroups of SO(4)

To perform the final step, consider a homomophism

$\displaystyle F \colon S^3 \times S^3 \to SO(4), \quad F(q_1, q_2) = q_1 \cdot q \cdot q_2^{-1}.$

Its kernel is equal to $\{(1,1), (-1,-1)\}$$\{(1,1), (-1,-1)\}$.

Finite subgroups of $S^3 \times S^3$$S^3 \times S^3$ can be determined by Goursat's lemma. This lemma says, that every finite subgroup of $S^3 \times S^3$$S^3 \times S^3$ is isomorphic to the fibre product $G \times_{Q} H$$G \times_{Q} H$, where $G$$G$ and $H$$H$ are finite subgroups of $S^3$$S^3$ and $Q$$Q$ is a common quotient of $G$$G$ and $H$$H$.

### 2.4 Finite fixed-point free subgroups of SO(4)

Not every finite subgroup of $SO(4)$$SO(4)$ act freely on $S^3$$S^3$. Following lemma gives necessary and sufficient condition for $F(q_1,q_2)$$F(q_1,q_2)$ to be fixed point free for $q_1, q_2 \in S^3$$q_1, q_2 \in S^3$.

Lemma 2.3 [Wolf2011]. Let $q_1, q_2$$q_1, q_2$ be unit quaternions, then $F(q_1,q_2)$$F(q_1,q_2)$ has a fixed point on $S^3$$S^3$ if, and only if, $q_1$$q_1$ is conjgate to $q_2$$q_2$ in $S^3$$S^3$.

Proof. This is a simple observation

$\displaystyle q_1 \cdot a \cdot q_2^{-1} = a \iff q_1 = a \cdot q_2 \cdot a^{-1}.$
$\square$$\square$

Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of $SO(4)$$SO(4)$ belongs to the following list

• cyclic group,
• generalised quaternion group $Q_{8k}$$Q_{8k}$,
• binary tetrahedral group $T^{\ast}$$T^{\ast}$,
• binary icosahedral group $I^{\ast}$$I^{\ast}$,
• groups $D_{2^k(2n+1)}$$D_{2^k(2n+1)}$, for $k \geq 2$$k \geq 2$ and $n \geq 1$$n \geq 1$, with presentation
$\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.$
• groups $P_{8 \cdot 3^{k}}'$$P_{8 \cdot 3^{k}}'$ defined by the following presentation
$\displaystyle \langle x,y,z | x^2 = (xy)^2 = y^2, zxz^{-1} = y, zyz^{-1} = xy, z^{3^k} = 1 \rangle,$
• direct product of any of the above group with a cyclic group of relatively prime order.

## 3 Milnor's contribution

In [Milnor1957] author proves the following theorem.

Theorem 3.1. If $G$$G$ is a finite group which admits a fixed-point free action on a sphere $S^{2n+1}$$S^{2n+1}$, then for every prime $p$$p$ every subgroup of $G$$G$ of order $2p$$2p$ is cyclic.

This theorem allowed him to choose all possible candidates, from the list compiled by Zassenhaus and Suzuki of groups with periodic cohomology, which could possibly act on $S^3$$S^3$. Apart from groups which admit a fixed point free representation in $SO(4)$$SO(4)$ he obtained the following family of groups.

$\displaystyle Q(8n,k,l) = \langle x,y,z \mid x^2 = (xy)^2=y^{2n}, z^{kl}=1, xzx^{-1} = z^{r}, yzy^{-1} = z^{-1} \rangle,$

where $8n,k,l$$8n,k,l$ are relatively prime integers and

$\displaystyle r \equiv -1 \pmod{k},$
$\displaystyle r \equiv 1 \pmod{l}.$

Groups $Q(8n,k,l)$$Q(8n,k,l)$ were excluded from the list of fundamental groups of $3$$3$-manifolds only after resolution of the Geometrization Conjecture.