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Almost all submaximal groups are paracompact and σ-discrete

O. Alas, I. Protasov, M. Tkačenko, V. Tkachuk, R. Wilson, I. Yaschenko (1998)

Fundamenta Mathematicae

We prove that any topological group of a non-measurable cardinality is hereditarily paracompact and strongly σ-discrete as soon as it is submaximal. Consequently, such a group is zero-dimensional. Examples of uncountable maximal separable spaces are constructed in ZFC.

Almost maximal topologies on groups

Yevhen Zelenyuk (2016)

Fundamenta Mathematicae

Let G be a countably infinite group. We show that for every finite absolute coretract S, there is a regular left invariant topology on G whose ultrafilter semigroup is isomorphic to S. As consequences we prove that (1) there is a right maximal idempotent in βG∖G which is not strongly right maximal, and (2) for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination (-,-,+), there is a corresponding regular almost maximal left invariant...

Almost periodic compactifications of group extensions

H. D. Junghenn, Paul Milnes (2002)

Czechoslovak Mathematical Journal

Let N and K be groups and let G be an extension of N by K . Given a property 𝒫 of group compactifications, one can ask whether there exist compactifications N ' and K ' of N and K such that the universal 𝒫 -compactification of G is canonically isomorphic to an extension of N ' by K ' . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties 𝒫 and then apply this result to the almost periodic and weakly almost periodic compactifications of G .

Amenable hyperbolic groups

Pierre-Emmanuel Caprace, Yves de Cornulier, Nicolas Monod, Romain Tessera (2015)

Journal of the European Mathematical Society

We give a complete characterization of the locally compact groups that are non elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semiregular trees acting doubly...

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