| 研究実績の概要 |
A subset of a group G is unconditionally closed in G if it is closed in every Hausdorff group topology on G. The family of all unconditionally closed subsets of G forms the family of closed subsets of a unique topology on G called its Markov topology. Similarly, a family of subsets of G which are closed in each precompact group topology on G coincides with the family of closed subsets of the so-called precompact Markov topology of G. We prove that every unconditionally closed subset of a free group is algebraic, thereby answering a problem of Markov for free groups. In modern terminology this means that Markov and Zariski topologies coincide for free groups. Moreover, we show that for non-commutative free groups, Markov topology differs from precompact Markov topology. This is accomplished by finding a sequence S in the free group F with two generators which converges to the identity in each precompact Hausdorff group topology on F (and thus, in the precompact Markov topology on F), yet there exists a Hausdorff group topology on F such that S does not converge to the identity with respect to this topology (and thus, S is not closed in the Markov topology of G). We also deduce from our results that the class of groups for which Markov and Zariski topologies coincide is not closed under taking quotients.
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