Conjugacy separability of HNN-extensions of abelian groups

1996 ◽  
Vol 67 (5) ◽  
pp. 353-359 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Abelian Groups ◽  
Hnn Extensions ◽  
Conjugacy Separability

Conjugacy Separability of Certain Polygonal Products

1996 ◽  
Vol 39 (3) ◽  
pp. 294-307 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Cyclic Subgroups ◽  
Conjugacy Separability ◽  
Conjugacy Separable

AbstractWe show that polygonal products of polycyclic-by-finite groups amalgamating central cyclic subgroups, with trivial intersections, are conjugacy separable. Thus polygonal products of finitely generated abelian groups amalgamating cyclic subgroups, with trivial intersections, are conjugacy separable. As a corollary of this, we obtain that the group A1 *〈a1〉A2 *〈a2〉 • • • *〈am-1〉Am is conjugacy separable for the abelian groups Ai.

scholarly journals On finite state automaton actions of HNN extensions of free abelian groups

2021 ◽  
Vol 13 (1) ◽  
pp. 180-188
Author(s):  
V. Prokhorchuk
Keyword(s):  
Finite Automaton ◽  
Sylow Subgroup ◽  
Abelian Groups ◽  
Finite State Automaton ◽  
Hnn Extensions ◽  
Finite State ◽  
Free Abelian Groups

HNN extensions of free abelian groups are considered. For arbitrary prime $p$ it is introduced a class of such extensions that act by finite automaton permutations over an alphabet $ \mathsf{X} $ of cardinality $p$ and belong to $p$-Sylow subgroup of the group of automaton permutations over such $ \mathsf{X} $. As a corollary it implies that all corresponding HNN extensions are residually $p$-finite.

Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

1999 ◽  
Vol 42 (3) ◽  
pp. 335-343 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Residual Finiteness ◽  
Necessary And Sufficient Condition ◽  
Residual Properties ◽  
Hnn Extensions ◽  
Subgroup Separability ◽  
Necessary And Sufficient

AbstractWe derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties ofHNN-extensions of nilpotent groups with cyclic associated subgroups.

Cyclic conjugacy separability and conjugacy separability of certain HNN extensions

2020 ◽  
Vol 48 (8) ◽  
pp. 3573-3589
Author(s):  
H. M. Lim ◽  
K. B. Wong ◽  
P. C. Wong
Keyword(s):  
Hnn Extensions ◽  
Conjugacy Separability

scholarly journals Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

2021 ◽  
pp. 1-26
Author(s):  
EDUARDO SILVA
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Global Constraints ◽  
Finitely Generated ◽  
Subshift Of Finite Type ◽  
Mixing Properties ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Periodic Configuration ◽  
The Right

Abstract For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

Residual finiteness, subgroup separability and conjugacy separability of certain HNN extensions

2012 ◽  
Vol 62 (5) ◽  
Author(s):  
K. Wong ◽  
P. Wong
Keyword(s):  
Finite Groups ◽  
Finite Subgroup ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Conjugacy Separability ◽  
Subgroup Separability ◽  
Conjugacy Separable

AbstractIn this note we shall give characterisations for HNN extensions of non-cyclic polycyclic-by-finite groups with normal infinite cyclic associated subgroups to be residually finite, subgroup separable and conjugacy separable.

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