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A property which ensures that a finitely generated hyper-(Abelian-by-finite) group is finite-by-nilpotent

Fares Gherbi, Nadir Trabelsi (2024)

Czechoslovak Mathematical Journal

Let 𝔐 be the class of groups satisfying the minimal condition on normal subgroups and let Ω be the class of groups of finite lower central depth, that is groups G such that γ i ( G ) = γ i + 1 ( G ) for some positive integer i . The main result states that if G is a finitely generated hyper-(Abelian-by-finite) group such that for every x ∈ G , there exists a normal subgroup H x of finite index in G satisfying 〈 x , x h 〉 ∈ 𝔐 Ω for every h ∈ H x , then G is finite-by-nilpotent. As a consequence of this result, we prove that a finitely generated hyper-(Abelian-by-finite)...

A reconstruction theorem for locally moving groups acting on completely metrizable spaces

Edmund Ben-Ami (2010)

Fundamenta Mathematicae

Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category...

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