Paradoxical subsets of hyperbolic and spherical discs
A subset E of the Euclidean plane \(E^ 2\) (respectively of the hyperbolic plane \(H^ 2\) or of the sphere \(S^ 2)\) is said to be (m,n)- paradoxical (with \(m\leq n)\) if E is nonempty and there are nonempty subsets \(C_ 1,...,C_ m\), \(D_ 1,...,D_ n\) of E and planar (resp. hyperbolic; spherical) isometries \(F_ 1,...,F_ m\); \(G_ 1,...,G_ n\) such that \(P_ 1=\{C_ i\}\), \(P_ 2=\{D_ j\}\) and \(P_ 3=\{F_ i(C_ i)\}\cup \{G_ j(D_ j)\}\) are each partitions of \(E^ 2\) (resp. of \(H^ 2\); \(S^ 2).\) The author completely answers the following question: (*) Given a disc D of radius r in \(H^ 2\) (or \(S^ 2)\) with \(0<r\leq \infty\), for which pairs (m,n), with \(m\leq n\), does there exist an (m,n)-paradoxical subset of D but not an (m-1,n)-paradoxical subset of D or an (m,n-1)-paradoxical subset of D? Namely: (1) in \(H^ 2:\) if \(0<r<\infty\) then (1,3) and (2,2); if \(r=\infty\) then (1,1). (2) in \(S^ 2:\) if \(0<r<\pi /2\) then (1,3) and (2,2); if \(\pi /2\leq r<\pi\) then (1,2); if \(r=\pi\) then (1,1). The problem goes back to \textit{S. Mazurkiewicz} and \textit{W. Sierpiński} [C. R. Acad. Sci. Paris 158, 618-619 (1914), JFM 45.0131.01], where (*) was answered for \(E^ 2\) with \(r=\infty\); namely (1,1). \textit{W. Just} [Bull. Pol. Acad. Sci., Math. 36, No.1/2, 1-3 (1988; Zbl 0683.51016)] and the author (in a paper to appear in Fund. Math.) have also completely answered (*) for \(E^ 2\) with \(0<r<\infty:\) the answer is (1,3) and (2,2). The constructions of the paradoxical subsets (which are all denumerable) are carried out by using two independent isometries (i.e. they generate a free subgroup of rank 2) and do not use AC.
scientific article; zbMATH DE number 4151246