A free group acting without fixed points on the rational unit sphere
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چکیده
We prove the existence of a free group of rotations of rank 2 which acts on the rational unit sphere without non-trivial fixed points. Introduction. The purpose of this paper is to prove that the group SO3(Q) of all proper orthogonal 3 × 3 matrices with rational entries has a free subgroup F2 of rank 2 such that for all w ∈ F2 different from the identity and for all ~r ∈ S2 ∩Q3 we have w(~r ) 6= ~r (Theorem 2). The question if such a group exists was raised by Professor J. Mycielski. Theorem 2 has the following corollary. The rational unit sphere S2 ∩Q3 (= {~r ∈ Q3 : |~r | = 1}) has all possible kinds of Banach–Tarski paradoxical decompositions, e.g. a partition into three sets A, B, and C such that A ≈ B ≈ C ≈ A ∪B ≈ B ∪ C ≈ C ∪A, where ≈ denotes congruence by a transformation of F2 (such a partition is called a Hausdorff decomposition). The proof of this corollary of Theorem 2 is well known (see e.g. [W, Cor. 4.12]). Moreover, since in this case the space S2 ∩Q3 is countable, the proof does not require the axiom of choice. A Hausdorff decomposition is not possible for the real sphere S2 (= {~r ∈ R3 : |~r | = 1}) relative to SO3(R) (= the group of all proper orthogonal matrices) since every rotation of S2 has fixed points (thus C ≈ A ∪ B cannot hold). However, it is possible if reflections are allowed (see [A] or [W, Theorem 4.16]). Other constructions of free subgroups of SO3(Q) are known. S. Świercz1991 Mathematics Subject Classification: Primary 20E05, 20H05, 20H20; Secondary 15A18, 51F20, 51F25. The author is greatful to Professor W. Takahashi for his encouragement and for much kind-hearted support, and also to Professor J. Mycielski for a suggestion of searching μ and ν which satisfy Theorem 2 and for some valuable comments.
منابع مشابه
Free groups of rotations acting without fixed points on the rational unit sphere
The purpose of this paper is to generalize an example of Sato of a free group of rotations of the Euclidean 3-space whose action on the rational unit sphere is fixed point free. Such groups are of interest, as they can be employed to construct paradoxical decompositions of spheres without assuming the Axiom of Choice.
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