Topological groups with fixed point on compacta property and potentially dense subsets of groups

设G是一个群，w是G与循环群Z（其生成元记为z）的自由积G*Z中的一个字。方程w=1的解集是G*Z的所有元素x的集合，使得w '= 1，其中w'是通过用x替换z在w中的所有出现而从w获得的字。群G的Zapriki（verbal）拓扑是G上所有方程w=1的解集在其中闭的最小拓扑，群G的子集在G中无条件闭，如果它在G的每个Hausdorff群拓扑中闭.群G的所有无条件闭子集族构成G上唯一拓扑的闭子集族，称之为G的Markov拓扑，群G的子群H嵌入G中，如果H的Markov拓扑是从G的Markov拓扑继承而来的子空间拓扑，则称H嵌入G中.群G的子群H称为嵌入G的Hausdorff，如果H上的每个Hausdorff群拓扑都可以扩张为G的一个Hausdorff群拓扑，使得原拓扑成为子群拓扑。我们证明了自由群的每个子群都是Zeroki和Markov嵌入的，另一方面，我们构造了一个有2个生成元的自由群的正规子群，它不嵌入Hausdorff.

Let G be a group, and let w be a word in the free product G*Z of G with the cyclic group Z (whose generator is denoted by z). The solution set of an equation w=1 is the set of all elements x of G*Z such that w'=1, where w' is the word obtained from w by replacing all occurencies of z in w with x. The Zariski (verbal) topology of a group G is the smallest topology on G in which solution set of all equations w=1 in G are closed.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.A subgroup H of a group G is Zariski (Markov) embedded in G if the Zariski (Markov) topology of H is the subspace topology it inherits from the Zariski (Markov) topology of G. A subgroup H of a group G is Hausdorff embedded in G if every Hausdorff group topology on H can be extended to a Hausdorff group topology of G in such a way that the original topology becomes a subgroup topology. We prove that every subgroup of a free group is both Zariski and Markov embedded in it. On the other hand, we construct a normal subgroup of a free group with 2 generators which is not Hausdorff embedded in it.