On Closed Subsets of Free Groups (original) (raw)
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Profinite Topologies in Free Products of Groups
International Journal of Algebra and Computation, 2004
Let [Formula: see text] be a nonempty class of finite groups closed under taking subgroups, quotients and extensions. We consider groups G endowed with their pro-[Formula: see text] topology, and say that G is 2-subgroup separable if whenever H and K are finitely generated closed subgroups of G, then the subset HK is closed. We prove that if the groups G1 and G2 are 2-subgroup separable, then so is their free product G1*G2. This extends a result to T. Coulbois. The proof uses actions of groups on abstract and profinite trees.
On Finitely Generated Subgroups of Free Products
Journal of the Australian Mathematical Society, 1971
If H is a subgroup of a group G we shall say that G is H-residually finite if for every element g in G, outside H, there is a subgroup of finite index in G, containing H and still avoiding g. (Then, according to the usual definition, G is residually finite if it is E-residually finite, where E is the identity subgroup). Definitions of other terms used below may be found in § 2 or in [6].
Some nonprojective subgroups of free topological groups
Proceedings of the American Mathematical Society, 1975
For the free topological group on an interval [a, b] a family of closed, locally path-connected subgroups is given such that each group is not projective and so not free topological. Simplicial methods are used, and the test for nonprojectivity is nonfreeness of the group of path components. Similar results are given for the abelian case.
Extending partial automorphisms and the profinite topology on free groups
Transactions of the American Mathematical Society, 1999
A class of structures C is said to have the extension property for partial automorphisms (EPPA) if, whenever C 1 and C 2 are structures in C, C 1 finite, C 1 ⊆ C 2 , and p 1 , p 2 ,. .. , pn are partial automorphisms of C 1 extending to automorphisms of C 2 , then there exist a finite structure C 3 in C and automorphisms α 1 , α 2 ,. .. , αn of C 3 extending the p i. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskiȋ stating that a finite product of finitely generated subgroups is closed for this topology.
Characterization of Bases of Subgroups of Free Topological Groups
Journal of the London Mathematical Society, 1983
It is shown here that if Y is a closed subspace of the free topological group FM (X) on a fc^-space X then FM {X) has a closed subgroup topologically isomorphic to FM (Y). Thus the problem of determining whether FM (Y) can be embedded in FM (X) is reduced to that of checking if FM (X) contains a closed copy of y. As an extension of the above result it is shown that if A y , A 2 ,..., A n are (not necessarily distinct) closed subspaces of FM (X), where X is a fc^-space, then FM {X) has a closed subgroup topologically isomorphic to FM (^ x A 2 x ... x A n).
A remark on free topological groups with no small subgroups
Journal of the Australian Mathematical Society, 1974
For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].
On the free profinite products of profinite groups with commuting subgroups
International Journal of Group Theory, 2016
In this paper we introduce the construction of free profinite products of profinite groups with commuting subgroups. We study a particular case: the proper free profinite products of profinite groups with commuting subgroups. We prove some conditions for a free profinite product of profinite groups with commuting subgroups to be proper. We derive some consequences. We also compute profinite completions of free products of (abstract) groups with commuting subgroups.
Metrizability of subgroups of free topological groups
Bulletin of the Australian Mathematical Society, 1986
It is shown that any sequential subgroup of a free topological group is either sequential of order ω1 or discrete. Hence any metrizable subgroup of a free topological group is discrete.
Publicacions Matemàtiques, 2003
Let V be a pseudovariety of finite groups such that free groups are residually V, and let ϕ: F (A) → F (B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extensionφ of ϕ between the pro-V completions of F (A) and F (B) is also injective. In particular, if V is extension-closed, this is the case if and only if ϕ(F (A)) and its pro-V closure in F (B) have the same rank. We examine a number of situations where the injectivity ofφ can be asserted, or at least decided, and we draw a few corollaries.