Reflection principle characterizing groups in which unconditionally closed sets are algebraic

We give a necessary and sufficient condition, in terms of a certain reflection principle, for every unconditionally closed subset of a group G to be algebraic. As a corollary, we prove that this is always the case when G is a direct product of an Abelian group with a direct product (sometimes also called a direct sum) of a family of countable groups. This is the widest class of groups known to date where the answer to the 63 years old problem of Markov turns out to be positive. We also prove that whether every unconditionally closed subset of G is algebraic or not is completely determined by countable subgroups of G. According to Markov [6], a subset S of a group G is called: (a) elementary algebraic if there exist an integer n > 0, a1, . . . , an ∈ G and ε1, . . . , εn ∈ {−1, 1} such that S = {x ∈ G : xa1xa2 . . . an−1x = an}, (b) algebraic if S is an intersection of finite unions of elementary algebraic subsets of G, (c) unconditionally closed if S is closed in every Hausdorff group topology of G. Since the family of all finite unions of elementary algebraic subsets of G is closed under finite unions and contains all finite sets, it is a base of closed sets of some T1 topology ZG on G, called the Zariski topology of G. (This topology is also known under the name verbal topology , see [1].) The family of all unconditionally closed subsets of G coincides with the family of closed subsets of a T1 topology MG on G, namely the infimum (taken in the lattice of all topologies on G) of all Hausdorff group topologies on G. We call MG the Markov topology of G. Note that (G,ZG) and (G,MG) are quasi-topological groups, i.e., the inversion and shifts are continuous. Fact 1. ZG ⊆ MG for every group G. Proof. An elementary algebraic subset of G must be closed in every Hausdorff group topology on G. In 1944 Markov [6] asked if the equality ZG = MG holds for every group G. He himself obtained a positive answer in case G is countable: Fact 2. (Markov’s theorem [6]) ZG = MG for every countable group G. Moreover, in the same manuscript [6] Markov attributes to Perel’man the fact that ZG = MG for every Abelian group G. To the best of our knowledge the proof of this fact has never appeared in print until [2]. (We offer an alternative self-contained proof of this result in Corollary 4.4.) A consistent example of a group G with ZG 6= MG was announced quite recently in [10]. Dipartimento di Matematica e Informatica, Unive rsità di Udine, Via delle Scienze 206, 33100 Udine, Italy; e-mail : [email protected] Graduate School of Science and Engineering, Division of Mathematics, Physics and Earth Sciences, Ehime University, Matsuyama 790-8577, Japan; e-mail : [email protected]