Fundamental groups of 3-dimensional spherical space forms

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Contents

1 Introduction

The purpose of this article is to describe fundamental groups of 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-dimensional spherical space forms. These are exactly groups which admit a fixed-point free representation in 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. All groups mentioned earlier belong to this list however there was also included a family of finite groups denoted by Q(8n,k,l). Question whether these groups act on 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) it is necessary to proceed in three steps:

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

2.1 Finite subgroups of SO(3)

To classify finite subgroups of SO(3) one has to analyse action of these groups on 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 denotes the order of the group, q denotes number of orbits with non-trivial isotropy and 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) is either

These groups are called polyhedral groups.

2.2 Finite subgroups of S^3

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

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

This action preserves 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) with kernel equal to \{\pm 1\}.

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

Finite subgroups of 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 is isomorphic to the fibre product G \times_{Q} H, where G and H are finite subgroups of S^3 and Q is a common quotient of G and H.

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

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

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

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

\displaystyle \langle x, y \mid x^{2^k} = y^{2n+1} = 1, xyx^{-1} = y^{-1} \rangle.
  • groups 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 is a finite group which admits a fixed-point free action on a sphere S^{2n+1}, then for every prime p every subgroup of G of order 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. Apart from groups which admit a fixed point free representation in 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 are relatively prime integers and

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

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

4 References

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