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

设G是一个群，设w是G与循环群Z（其生成源记为Z）的自由积G*Z中的一个词。方程w=1的解集是G*Z中使w‘=1的所有元素x的集合，其中w’是用x代替w中所有出现的Z而从w中得到的词。群G的Zariski （verbal）拓扑是G上所有方程w=1的解集闭合的最小拓扑。如果群G的子集在G上的每一个Hausdorff群拓扑上都是闭的，那么它在G上是无条件闭的。G的所有无条件闭子集的族构成了G上的一个唯一拓扑的闭子集族，称为它的Markov拓扑。群G的子群H是嵌入在G中的Zariski (Markov)，如果H的Zariski （Markov）拓扑是继承自G的Zariski （Markov）拓扑的子空间拓扑，则群G的子群H是嵌入在G中的Hausdorff，如果H上的每个Hausdorff群拓扑可以扩展到G的Hausdorff群拓扑，从而使原始拓扑成为子群拓扑。证明了自由群的每一子群都同时嵌入Zariski和Markov。另一方面，我们构造了一个自由群的正规子群，其中嵌入了2个非Hausdorff生成器。

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.