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Growth functions for some uniformly amenable groups

100%
Dronka J., Wajnryb B., Witowicz P., Orzechowski K.
Open Mathematics
|
2017
|
tom 15
|
nr 1
502-507
EN
We present a simple constructive proof of the fact that every abelian discrete group is uniformly amenable. We improve the growth function obtained earlier and find the optimal growth function in a particular case. We also compute a growth function for some non-abelian uniformly amenable group.
2
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When the intrinsic algebraic entropy is not really intrinsic

100%
Goldsmith B., Salce L.
Topological Algebra and its Applications
|
2015
|
tom 3
|
nr 1
EN
The intrinsic algebraic entropy ent(ɸ) of an endomorphism ɸ of an Abelian group G can be computed using fully inert subgroups of ɸ-invariant sections of G, instead of the whole family of ɸ-inert subgroups. For a class of groups containing the groups of finite rank, aswell as those groupswhich are trajectories of finitely generated subgroups, it is proved that only fully inert subgroups of the group itself are needed to comput ent(ɸ). Examples show how the situation may be quite different outside of this class.
3
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Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

88%
Goldsmith B., Gong K.
Topological Algebra and its Applications
|
2015
|
tom 3
|
nr 1
EN
Necessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.
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