ar X iv : m at h / 06 03 51 3 v 1 [ m at h . G R ] 2 1 M ar 2 00 6 Hereditarily non - topologizable groups ∗

A group G is non-topologizable if the only Hausdorff group topology that G admits is the discrete one. Is there an infinite group G such that H/N is non-topologizable for every subgroup H ≤ G and every normal subgroup N ⊳ H? We show that a solution of this essentially group theoretic question provides a solution to the problem of c-compactness. Following Ol'shanskiˇı, we say that a group G is non-topologizable if the only Hausdorff group topology that G admits is the discrete one. In 1944, Markov asked whether infinite non-topologizable groups exist (cf. [8]). Markov's problem was solved in 1980, independently, by Ol'shanskiˇı and Shelah, who constructed infinite non-topologizable groups ([9] and [12]). Although Ol'shanskiˇı's example is countable and periodic, Klyachko and Trofimov showed that a non-topologizable group need not satisfy either of these properties. Theorem 1. ([4]) There exists a torsion-free finitely generated non-topologizable group. Thus, there exists a torsion-free non-topologizable group of any cardinality. (The second statement is obtained from the first one using Löwenheim-Skolem theorem.) In a subsequent paper, Trofimov proved that every group embeds into a non-topologizable group of the same cardinality (cf. [14, Thm. 3]). Markov himself obtained a criterion of non-topologizability for countable groups with a strong algebraic geometric flavour (Theorem 2 below), whose most elegant proof was given by Ze-lenyuk and Protasov, more than half a century later (cf.