Free subgroups of free complete products
The free complete product \(G\) of groups \(G_i\) is very much like the standard free product except that one allows, in addition to the standard finite words, arbitrarily large linearly ordered ``words'', subject only to the condition that, to each word, each group \(G_i\) contributes only finitely many letters. This finiteness condition allows a natural projection into each standard free product of finitely many of the \(G_i\)'s. Equivalence of words is determined by these projections. Multiplication is by concatenation. If each \(G_i\) is isomorphic with the group of integers, the standard results are the following: (1) If the indexing set is infinite, then \(G\) is an uncountable group that is not free but has very large free subgroups (Zastrow, Cannon-Conner, Eda). (2) The group \(G\) has infinitely generated free retracts if (Shelah-Strüngmann and Eda-Shelah) and only if (many authors) the indexing set is uncountable. Conner and Eda give the appropriate analogues of these results when the integers are replaced by arbitrary groups. They generalize the free subgroup constructions of Zastrow and Cannon-Conner and the retract theorems of Shelah-Strüngmann.
scientific article; zbMATH DE number 1763553