The failure of the uncountable non-commutative Specker Phenomenon

Higman proved in 1952 that every free group is non-commutatively slender, that is, for a free group G and for a homomorphism h from the free complete product x_ωℤ of countably many copies of ℤ into G, there exists a finite subset F ⊆ ω and a homomorphism h̄: *_Fℤ → G such that h = h̄p_F, where p_F is the natural map from x_ωℤ into *_Fℤ. Because of the corresponding phenomenon for abelian groups this is called the non-commutative Specker Phenomenon. In the present paper we shall show that Higman's result fails if one passes from countable to uncountable and thereby answer a question posed by K. Eda. In particular, we will see that, for an uncountable cardinal λ and for non-trivial groups G_x (α ∈ λ), there are 2^2λ homomorphisms from the free complete product of the groups G_x into the integers.