Free and non-free subgroups of the fundamental group of the Hawaiian earrings.
The Hawaiian earring \(H=C_1\cup C_2\cup\cdots\) is the compact planar set that is the union of the circles \(C_n\) of radius \(1/n\) and center \((0,-1/n)\). The fundamental group \(G=\pi_1(H,0)\) of \(H\) is an uncountable group that is locally free but not free. Considerable effort has been expended in recent years to show that the group \(G\) nevertheless deserves to be considered free in some generalized sense and to determine which of its many subgroups are free or not free in the classical sense. This paper adds to the pantheon of examples. Most interesting are the following: (1) The first known example of a countable subgroup of \(G\) that is not free. (2) A subgroup that is free, contains the standard ``generators'' (the classes of the simple closed curves in \(H)\), but for which those generators cannot be extended to a free basis. (3) An uncountable free subgroup that differs substantially from all known free subgroups. (4) Examples showing how little one needs to change the description to pass from a free subgroup to a subgroup that is not free.
scientific article; zbMATH DE number 1862498