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

设G是一个群，w是G的自由乘积G*Z中的一个词，其循环群Z(其生成元记为z)。方程w=1的解集是G*Z的所有元素x的集合，使得w‘=1，群G的Zariski(文字)拓扑是G上所有方程的解集都是闭的最小拓扑。如果群G的子集在G上的每个Hausdorff群拓扑中是闭的，则它在G中是无条件闭的。G的所有无条件闭子集族形成了G上唯一拓扑的闭子集族，称为它的马尔可夫拓扑。群G的子群H是嵌入在G中的Zariski(马尔可夫)拓扑群G的子群H是嵌入在G中的Hausdorff子群，如果H上的每个Hausdorff群拓扑都能扩张为G的Hausdorff群拓扑，从而使原拓扑成为子群拓扑。我们证明了自由群的每个子群都嵌入了Zariski和马氏子群。另一方面，我们构造了一个自由群的正规子群，它有两个非Hausdorff嵌入的生成元。

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.