• Title/Summary/Keyword: polygonal products

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SEPARABILITY PROPERTIES OF CERTAIN POLYGONAL PRODUCTS OF GROUPS

  • Kim, Goan-Su;Tang, C.Y.
    • Journal of the Korean Mathematical Society
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    • v.39 no.3
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    • pp.461-494
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    • 2002
  • Let G = E${\ast}_{A}F$, where A is a finitely generated abelian subgroup. We prove a criterion for G to be {A}-double coset separable. Applying this result, we show that polygonal products of central subgroup separable groups, amalgamating trivial intersecting central subgroups, are double coset separable relative to certain central subgroups of their vertex groups. Finally we show that such polygonal products are conjugacy separable. It follows that polygonal products of polycyclic-by-finite groups, amalgamating trivial intersecting central subgroups, are conjugacy separable.

OUTER AUTOMORPHISM GROUPS OF CERTAIN POLYGONAL PRODUCTS OF GROUPS

  • Kim, Goan-Su
    • Bulletin of the Korean Mathematical Society
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    • v.45 no.1
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    • pp.45-52
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    • 2008
  • We show that certain polygonal products of any four groups, amalgamating central subgroups with trivial intersections, have Property E. Using this result, we derive that outer automorphism groups of polygonal products of four polycyclic-by-finite groups, amalgamating central subgroups with trivial intersections, are residually finite.

POLYGONAL PRODUCTS OF RESIDUALLY FINITE GROUPS

  • Wong, Kok-Bin;Wong, Peng-Choon
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.1
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    • pp.61-71
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    • 2007
  • A group G is called cyclic subgroup separable for the cyclic subgroup H if for each $x\;{\in}\;G{\backslash}H$, there exists a normal subgroup N of finite index in G such that $x\;{\not\in}\;HN$. Clearly a cyclic subgroup separable group is residually finite. In this note we show that certain polygonal products of cyclic subgroup separable groups amalgamating normal subgroups are again cyclic subgroup separable. We then apply our results to polygonal products of polycyclic-by-finite groups and free-by-finite groups.

OUTER AUTOMORPHISM GROUPS OF POLYGONAL PRODUCTS OF CERTAIN CONJUGACY SEPARABLE GROUPS

  • Kim, Goan-Su;Tang, Chi Yu
    • Journal of the Korean Mathematical Society
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    • v.45 no.6
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    • pp.1741-1752
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    • 2008
  • Grossman [7] showed that certain cyclically pinched 1-relator groups have residually finite outer automorphism groups. In this paper we prove that tree products of finitely generated free groups amalgamating maximal cyclic subgroups have residually finite outer automorphism groups. We also prove that polygonal products of finitely generated central subgroup separable groups amalgamating trivial intersecting central subgroups have residually finite outer automorphism groups.

CONJUGACY SEPARABILITY OF FREE PRODUCTS WITH AMALGAMATION

  • Kim, Goan-Su
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.521-530
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    • 1997
  • We first prove a criterion for the conjugacy separability of free products with amalgamation where the amalgamated subgroup is not necessarily cyclic. Applying this result, we show that free products of finite number of polycyclic-by-finite groups with central amalgamation are conjugacy separable. We also show that polygonal products of polycyclic-by-finite groups, amalgamating central cyclic subgroups with trivial intersections, are conjugacy separable.

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