• Title/Summary/Keyword: residual finiteness

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ON THE RESIDUAL FINITENESS OF FUNDAMENTAL GROUPS OF GRAPHS OF CERTAIN GROUPS

  • Kim, Goansu
    • Journal of the Korean Mathematical Society
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    • v.41 no.5
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    • pp.913-920
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    • 2004
  • We give a characterization for fundamental groups of graphs of groups amalgamating cyclic edge subgroups to be cyclic subgroup separable if each pair of edge subgroups has a non-trivial intersection. We show that fundamental groups of graphs of abelian groups amalgamating cyclic edge subgroups are cyclic subgroup separable, hence residually finite, if each edge subgroup is isolated in its containing vertex group.

CYCLIC SUBGROUP SEPARABILITY OF HNN EXTENSIONS

  • Kim, Goansu
    • Bulletin of the Korean Mathematical Society
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    • v.30 no.2
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    • pp.285-293
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    • 1993
  • In [4], Baumslag and Tretkoff proved a residual finiteness criterion for HNN extensions (Theorem 1.2, below). This result has been used extensively in the study of the residual finiteness of HNN extensions. Note that every one-relator group can be embedded in a one-relator group whose relator has zero exponent sum on a generator, and the latter group can be considered as an HNN extension. Hence the properties of an HNN extension play an important role in the study of one-relator groups [3], [2]. In this paper we prove a criterion for HNN extensions to be .pi.$_{c}$(Theorem 2.2). Moreover, we can prove that certain one-relator groups, known to be residually finite, are actually .pi.$_{c}$. It was known by Mostowski [10] that the word problem is solvable for finitely presented, residually finite groups. In the same way, the power problem is solvable for finitely presented .pi.$_{c}$ groups. Another application of subgroup separability with respect to special subgroups was mentioned by Thurston [12, Problem 15].m 15].

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RESIDUAL FINITENESS AND ABELIAN SUBGROUP SEPARABILITY OF SOME HIGH DIMENSIONAL GRAPH MANIFOLDS

  • Kim, Raeyong
    • Korean Journal of Mathematics
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    • v.29 no.3
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    • pp.603-612
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    • 2021
  • We generalize 3-manifolds supporting non-positively curved metric to construct manifolds which have the following properties : (1) They are not locally CAT(0). (2) Their fundamental groups are residually finite. (3) They have subgroup separability for some abelian subgroups.

Residual P-Finiteness of Certain Generalized Free Products of Nilpotent Groups

  • Kim, Goan-Su;Lee, Young-Mi;McCarron, James
    • Kyungpook Mathematical Journal
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    • v.48 no.3
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    • pp.495-502
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    • 2008
  • We show that free products of finitely generated and residually p-finite nilpotent groups, amalgamating p-closed central subgroups are residually p-finite. As a consequence, we are able to show that generalized free products of residually p-finite abelian groups are residually p-finite if the amalgamated subgroup is closed in the pro-p topology on each of the factors.

RESIDUAL p-FINITENESS OF CERTAIN HNN EXTENSIONS OF FREE ABELIAN GROUPS OF FINITE RANK

  • Chiew Khiam Tang;Peng Choon Wong
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.3
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    • pp.785-796
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    • 2024
  • Let p be a prime. A group G is said to be residually p-finite if for each non-trivial element x of G, there exists a normal subgroup N of index a power of p in G such that x is not in N. In this note we shall prove that certain HNN extensions of free abelian groups of finite rank are residually p-finite. In addition some of these HNN extensions are subgroup separable. Characterisations for certain one-relator groups and similar groups including the Baumslag-Solitar groups to be residually p-finite are proved.