The failure of the uncountable non-commutative Specker Phenomenon

TL;DR

This paper demonstrates that the non-commutative Specker Phenomenon, valid for countable free products, fails in the uncountable case, showing the existence of a vast number of homomorphisms in that setting.

Contribution

It proves that Higman's non-commutative Specker Phenomenon does not extend to uncountable free products, revealing a fundamental difference in their algebraic structure.

Findings

Fails for uncountable free products

Existence of 2^{2^lambda} homomorphisms

Contrasts with countable case

Abstract

Higman proved in 1952 that every free group is non-commutatively slender, this is to say that if G is a free group and h is a homomorphism from the countable complete free product (X_omega Z) to G, then there exists a finite subset F of omega and a homomorphism h:*_{i in F} Z --> G such that h=h rho_F, where rho_F is the natural map from (X_{i in omega})Z to *_{i in F}Z . Corresponding to the abelian case this phenomenon was called the non-commutative Specker Phenomenon. In this paper we show that Higman's result fails if one passes from countable to uncountable. In particular, we show that for non-trivial groups G_alpha (alpha in lambda) and uncountable cardinal lambda there are 2^{2^lambda} homomorphisms from the complete free product of the G_alpha 's to the ring of integers.