Countably compact groups having minimal infinite powers

Abstract:

We answer the question, raised more than thirty years ago by Dikranjan and Shakhmatov [Comp. Rend. Acad. Sci. Bulg., 41 (1990), pp. 13-15] and Dikranjan and Shakhmatov [Trans. Amer. Math. Soc. 335 (1993), pp. 775-790], on whether the power $G^\omega$ of a countably compact minimal Abelian group $G$ is minimal, by showing that the negative answer is equivalent to the existence of measurable cardinals. The proof is carried out in the larger class of sequentially complete groups. We characterize the sequentially complete minimal Abelian groups $G$ such that $G^\omega$ is minimal – these are exactly those $G$ that contain the connected component of their completion. This naturally leads to the next step, namely, a better understanding of the structure of the sequentially complete minimal Abelian groups, and in particular, their connected components which turns out to depend on the existence of Ulam measurable cardinals. More specifically, all connected sequentially complete minimal Abelian groups are compact if Ulam measurable cardinals do not exist. On the other hand, for every Ulam measurable cardinal $\sigma$ we build a non-compact torsion-free connected minimal $\omega$-bounded Abelian group of weight $\sigma$, thereby showing that the Ulam measurable cardinals are precisely the weights of non-compact sequentially complete connected minimal Abelian groups.