Some Subgroups of S03(IR) Isomorphic to the Free Product (Z/2Z) * (Z/3Z)
Kenzi Sat · 2016
The answer depends only on the value c = cos 0. This problem was introduced by Hausdorff, who showed that the answer is yes when c is transcendental [6, pp. 469473], [7], [11]. (Without Hausdorff's result, it is possible to show that the free product (Z/2Z) * (Z/3Z) is isomorphic to a subgroup of S 03(R) [2].) As a somewhat surprising application, one can use Hausdorff's result to prove the (Hausdorff-)BanachTarski theorem-it asserts that any two bounded sets with nonempty interiors in 3-dimensional Euclidean space can be partitioned into an equal number of pieces in such a way that corresponding pieces are congruent [11], [13, Theorem 3.11]by using the pair *44)4 and 0*4)~4r , which generates the free product Z *Z [8, p. 195], [13, p. 16]. (It is possible to use other pairs, for example, 4'4)4'4)4')-1' and q-li5f4r4)q [12, p. 80], 9 0*0*-' and 0*4-'0it [5], or 04V44) and 4)1-'41-' [3], [4], [10, p. 248], and so on.) The following cases of the problem have already been settled: