Fundamental groups of 3-dimensional spherical space forms
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1 Introduction
The purpose of this article is to describe fundamental groups of -dimensional spherical space forms. For the historical context refer to this article. Today we know the whole list of groups which appear as fundamental groups of -dimensional spherical space forms. These are exactly groups which admit a fixed-point free representation in . However in 1950's Milnor in [Milnor1957] provided a list of finite groups which could act freely but not linearly on . According to HAMBLETON it was not known whether groups from the Milnor's list can indeed act on until Perelman's resolution of the geometrization conjecture. The exposition is based on [Wolf2011].
2 Finite subgroups of SO(4)
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers to determine its finite groups.
2.1 Finite subgroups of SO(3)
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and 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 is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of S^3
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of SO(4)
To perform the final step, consider a homomophism
Its kernel is equal to .
Finite subgroups of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for to be fixed point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- direct product of any of the above group with a cyclic group of relatively prime order.
3 Milnor's contribution
4 References
- [Milnor1957] J. Milnor, Groups which act on without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of SO(4)
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers to determine its finite groups.
2.1 Finite subgroups of SO(3)
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and 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 is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of S^3
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of SO(4)
To perform the final step, consider a homomophism
Its kernel is equal to .
Finite subgroups of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for to be fixed point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- direct product of any of the above group with a cyclic group of relatively prime order.
3 Milnor's contribution
4 References
- [Milnor1957] J. Milnor, Groups which act on without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of SO(4)
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers to determine its finite groups.
2.1 Finite subgroups of SO(3)
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and 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 is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of S^3
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of SO(4)
To perform the final step, consider a homomophism
Its kernel is equal to .
Finite subgroups of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for to be fixed point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- direct product of any of the above group with a cyclic group of relatively prime order.
3 Milnor's contribution
4 References
- [Milnor1957] J. Milnor, Groups which act on without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219
2 Finite subgroups of SO(4)
To determine finite subgroups of it is necessary to proceed in three steps:
- determine finite subgroups of ,
- use the covering map to determine finite subgroups of ,
- use the fact that doubly covers to determine its finite groups.
2.1 Finite subgroups of SO(3)
First step in the determination of the groups from first family is the classification of finite subgroups of . This is done by analysing the action of this group on . From Riemann-Hurwitz formula we obtain an equation
where denotes the order of the group, denotes number of orbits with non-trivial isotropy and 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 is either cyclic, dihedral, tetrahedral, octahedral or icosahedral.
2.2 Finite subgroups of S^3
Let denote the algebra of quaternions and treat as a subset of of quaternions of norm . Consider an action of on by conjugation
This action preserves , so it induces an action on the set of imaginary quaternions which preserves the norm. Therefore this action yields a representation with kernel equal to .
If is a finite subgroup of , then let . If , then, since is the only element of of order , and are of odd order. Therefore is cyclic of odd order. If , then is an extension of the form
Theorem 2.2 [Wolf2011]. Every finite subgroup of is either cyclic group, binary dihedral group, binary tetrahedral group, binary octahedral group or binary icosahedralhedral group.
2.3 Finite subgroups of SO(4)
To perform the final step, consider a homomophism
Its kernel is equal to .
Finite subgroups of can be determined by Goursat's lemma. This lemma says, that every finite subgroup of is isomorphic to the fibre product , where and are finite subgroups of and is a common quotient of and .
2.4 Finite fixed-point free subgroups of SO(4)
Not every finite subgroup of act freely on . Following lemma gives necessary and sufficient condition for to be fixed point free for .
Lemma 2.3 [Wolf2011]. Let be unit quaternions, then has a fixed point on if, and only if, is conjgate to in .
Proof. This is a simple observation
Theorem 2.4 [Wolf2011]. Finite fixed-point free subgroup of belongs to the following list
- cyclic group,
- generalised quaternion group ,
- binary tetrahedral group ,
- binary icosahedral group ,
- groups , for and , with presentation
- groups defined by the following presentation
- direct product of any of the above group with a cyclic group of relatively prime order.
3 Milnor's contribution
4 References
- [Milnor1957] J. Milnor, Groups which act on without fixed points, Amer. J. Math. 79 (1957), 623–630. MR0090056 (19,761d)
- [Wolf2011] J. A. Wolf, Spaces of constant curvature, AMS Chelsea Publishing, Providence, RI, 2011. MR2742530 (2011j:53001) Zbl 05830219