The failure of the uncountable non-commutative Specker phenomenon (Q2769825)

The free complete product \(G\) of groups \(G_i\) is a generalization of the free product. It considers words \(W\) on the alphabet \(\bigcup G_i\), where a word is a linearly-ordered set of letters of arbitrary length and linear-order type, subject to the restriction that \(W\) contain only finitely many letters from each \(G_i\). Words are equivalent if all of their natural projections into finite free products of the groups \(G_i\) are equivalent. Multiplication is by concatenation. There are natural projections \(G\to G_{i_1}*\cdots*G_{i_k}\) for every finite collection \(i_1,\dots,i_k\) of indices.NEWLINENEWLINENEWLINELet \(G\) by the free complete product of countably many copies \(G_i\approx\mathbb{Z}\) of the integers \(\mathbb{Z}\), and let \(F\) be any free group. In 1952, Higman showed that any homomorphism from \(G\) into \(F\) factors through some one of the natural projections \(G\to G_{i_1}*\cdots*G_{i_k}\). For historical reasons, this result is called the non-commutative Specker phenomenon.NEWLINENEWLINENEWLINEThe authors show that this result fails for the free complete product \(G\) of uncountably many copies of \(\mathbb{Z}\). In fact, they show that there are more homomorphisms from \(G\) into \(\mathbb{Z}\) than would be allowed by the Specker phenomenon.