# Fundamental groups of 3-dimensional spherical space forms

## 1 Introduction

The purpose of this article is to describe fundamental groups of $3$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}3$-dimensional spherical space forms. For the historical context refer to this article. Today we know the whole list of groups which arise as fundamental groups of $3$$3$-dimensional spherical space forms. These are exactly groups which admit a fixed-point free representation in $SO(4)$$SO(4)$. In 1950's Milnor in [Milnor1957] provided a list of all finite groups which could possibly act freely but not necessarilly linearly on $S^3$$S^3$. All groups mentioned earlier belong to this list however there was also included a family of finite groups denoted by $Q(8n,k,l)$$Q(8n,k,l)$. Question whether these groups act on $S^3$$S^3$ remained unsolved until the proof of the Geometrization Conjecture was finished by Perelman. The exposition of the first part is based on [Wolf2011].

## 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}$$Q_{8k}$,
• binary octahedral group $O^{\ast}$$O^{\ast}$,
• 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.