Free products of topological groups with a closed subgroup amalgamated

Nickolas, Peter

doi:10.1017/s1446788700027580

Cited by 4 publications

(4 citation statements)

References 8 publications

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“…The following theorem shows one of the principal differences between the groups F(X) and A(X) (see [11,14,18] …”

Section: Theorem 1 1 For An Arbitrary Space X the Family {O 2 (Dmentioning

confidence: 99%

Local compactness in free topological groups

Nickolas¹,

Tkachenko²

2003

Bull. Austral. Math. Soc.

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We show that the subspace A n (X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n € u> if and only if A2(X) is locally compact if and only if ^( X ) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace F n (X) of the free topological group F(X) is locally compact for each n G w if and only if Fn(X) is locally compact if and only if F n (X) has pointwise countable type for each n e w i f and only if F^{X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that A n {X) has pointwise countable type for each n € w if and only if A2(X) has pointwise countable type if and only if •FM-Y) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that Fi(X) is locally compact if and only if F3(X) is locally compact, and that Fz(X) has pointwise countable type if and only if F$(X) has pointwise countable type.

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“…The following theorem shows one of the principal differences between the groups F(X) and A(X) (see [11,14,18] …”

Section: Theorem 1 1 For An Arbitrary Space X the Family {O 2 (Dmentioning

confidence: 99%

Local compactness in free topological groups

Nickolas¹,

Tkachenko²

2003

Bull. Austral. Math. Soc.

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“…that the composites Sf, S'f, (pr)/and (Pr)/ are ^-continuous. The ^-continuity of the first three is clear, while (Pr)/ sends a n (6h-»/(a, 6)), which is ^-continuous by the exponential law (10). Conversely, let g:i(A) -» O b e a fc-morphism over H. Its adjoint g:Ax H B ->C is defined algebraically by g(a,b) = ((Pr)(g(a))(b), and so is fc-continuous.…”

Section: E T /mentioning

confidence: 98%

“…The question of when a. [, a 2 are homeomorphisms into is more delicate; [10] gives a positive answer in the case A y , A 2 are & w -groups and a,, a 2 are closed inclusions. COROLLARY 8.…”

Section: Theorem 7 Let Ft: B -* Hbea Map In Kgp Which Is Topologicalmentioning

confidence: 99%

Lifting amalgamated sums and other colimits of groups and topological groups

Brown

Heath

1987

Math. Proc. Camb. Phil. Soc.

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Suppose a group H is given as a free product with amalgamationdetermined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.

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“…In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).…”

Section: Introductionmentioning

confidence: 99%

Local invariance of free topological groups

Khan

Morris

Nickolas

1986

Proceedings of the Edinburgh Mathematical Society

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In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)

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Free products of topological groups with a closed subgroup amalgamated

Cited by 4 publications

References 8 publications

Local compactness in free topological groups

Local compactness in free topological groups

Lifting amalgamated sums and other colimits of groups and topological groups

Local invariance of free topological groups

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