On the Conjugacy Separability of Certain Graphs of Groups

Conjugacy separability and outer automorphism groups of certain HNN extensions

“…This is done in Theorem 3.4. This result extends the result of Raptis, Talelli and Varsos [18,Theorem A]. From this, we are able to use the results of Wong and Wong [22] to give characterizations for these HNN extensions to be conjugacy separable in Theorem 3.5.…”

“…Then we prove the equivalence of residually finiteness and conjugacy separability in these HNN extensions (Theorem 3.4). This extends the result of Raptis, Talelli and Varsos [18,Theorem A]. In Section 5, we show that every conjugating endomorphism for these HNN extensions is also an inner automorphism (Theorem 5.1).…”

“…Hence by Theorem 3.3, G is conjugacy separable. 2 Note that Theorem 3.4 generalizes Theorem A of [18]. Now by Theorem 3.3 of [22] and Theorem 3.4, we have the following characterization.…”

“…However Kim and Tang had characterized the conjugacy separability of HNN extensions of finitely generated abelian groups with cyclic associated subgroups in [14] and they gave a criterion for the conjugacy separability in HNN extensions with cyclic associated subgroups in [15]. More recently Raptis, Talelli and Varsos in [18] proved the equivalence of residual finiteness and conjugacy separability in HNN extensions of finitely generated abelian groups and certain HNN extensions of finitely generated torsion free nilpotent groups. For completeness we state the following definitions.…”

Cyclic Subgroup Separability of Certain Graph Products of Subgroup Separable Groups

“…Recently, Zhou and Kim [31,32] had studied the class-preserving automorphisms of certain groups. Raptis, Talelli and Varsos [21] showed that conjugacy separability and residually finiteness are equivalent in certain HNN extensions (see also [26,27] for similar results).…”