Group actions on disks
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1 History
Floyd and Richardson [Floyd&Richardson1959] have constructed for the first time a smooth fixed point free action of on a disk for , the alternating group on five letters (see [Bredon1972, pp. 55-58] for a transparent description of the construction). Later, Greever [Greever1960]has described plenty of finite solvable groups not of prime power order, which can act smoothly on disks without fixed points. Then Oliver has answered completely the question of which compact Lie groups admit smooth fixed point free actions on disks (see [Oliver1975] and [Oliver1976]).
2 Oliver number
Proposition 1.1 [Oliver1975][Oliver1977][Oliver1978]. For a finite group not of prime power order, the following holds:
- if and only if there does not exist a sequence of normal subgroups such that is a -group, is a -group, and is cyclic for two (possibly equal) primes and .
3 Oliver group
In connection with the work on smooth one fixed point actions on spheres, Laitinen and Morimoto [Laitinen&Morimoto1998] have introduced the notion of Oliver group.
Definition 2.1. A finite group not of prime power order is called an Oliver group if .
Examples of finite Oliver groups include:
- finite nilpotent (in particular, abelian) groups with three or more non-cyclic Sylow subgroups (e.g. for three distinct primes , , and ).
- the groups and of order 72 (which are solvable but not nilpotent).
- finite non-solvable (in particular, non-trivial perfect) groups (e.g. , for ).
4 Solution
Theorem 3.1 [Oliver1975]. Let be a finite group not of prime power order. Then the following three conditions are equivalent.
- There exists a smooth fixed point free action of on some disk.
- The Actions on disks without fixed points#Oliver number .
Theorem 3.2 [Oliver1976]. Let be a compact Lie group with non-abelian identity connected component . Then there exist smooth fixed point free action of on some disk.
Corollary 3.3. A compact Lie group has a smooth fixed point free action on soome disk if and only if is non-abelian or .
5 References
- [Bredon1972] G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR0413144 (54 #1265) Zbl 0484.57001
- [Floyd&Richardson1959] E. E. Floyd and R. W. Richardson, An action of a finite group on an -cell without stationary points. , Bull. Amer. Math. Soc. 65 (1959), 73–76. MR0100848 (20 #7276) Zbl 0088.15302
- [Greever1960] J. Greever, Stationary points for finite transformation groups, Duke Math. J 27 (1960), 163–170. MR0110094 (22 #977) Zbl 0113.16505
- [Laitinen&Morimoto1998] E. Laitinen and M. Morimoto, Finite groups with smooth one fixed point actions on spheres, Forum Math. 10 (1998), no.4, 479–520. MR1631012 (99k:57078) Zbl 0905.57023
- [Oliver1975] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR0375361 (51 #11556) Zbl 0304.57020
- [Oliver1976] R. Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no.1, 79–96. MR0423390 (54 #11369) Zbl 0334.57023
- [Oliver1977] R. Oliver, -actions on disks and permutation representations. II, Math. Z. 157 (1977), no.3, 237–263. MR0646085 (58 #31126) Zbl 0386.20002
- [Oliver1978] R. Oliver, -actions on disks and permutation representations, J. Algebra 50 (1978), no.1, 44–62. MR0501044 (58 #18508) Zbl 0386.20002