Open sets satisfying systems of congruences (Q2772966)

The author generalizes arguments of Hausdorff who proved that a sphere with countably many points removed can be partitioned into three pieces \(A\), \(B\), \(C\) such that the sets \(A\), \(B\), \(C\), \({B\cup C}\) are mutually congruent. In the paper under review a congruence is specified by a finite system of variables \(A_1\), \dots, \(A_r\) and two sets \(L,R\subseteq\{1,\dots,r\}\) and formally it is written as \(\bigcup_{k\in L}A_k\cong\bigcup_{k\in R}A_k\). If \(G\) is a group acting on a set \(X\) and a system of congruences is given by pairs \(L_i,R_i\subseteq\{1,\dots,r\}\) for \(i\leq m\), then a sequence of pairwise disjoint sets \(A_k\subseteq X\), \(k\leq r\), is said to satisfy the system of congruences if for each \(i\leq m\) there is \(\sigma_i\in G\) such that \(\sigma_i(\bigcup_{k\in L_i}A_k)= \bigcup_{k\in R_i}A_k\). The author considers the problem which systems of congruences can be satisfied using open subsets of the sphere or related spaces. These open sets cannot form a partition of the sphere but they can cover ``almost all'' of the space. He pays more attention to the case of the sphere acted on by a free group of rotations than to the case of the sphere with its entire group of isometries. The reason is that these results often can be generalized to a wide variety of other suitable spaces.