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Free actions of free groups on countable structures and property (T)

David M. Evans, Todor Tsankov (2016)

Fundamenta Mathematicae

We show that if G is a non-archimedean, Roelcke precompact Polish group, then G has Kazhdan's property (T). Moreover, if G has a smallest open subgroup of finite index, then G has a finite Kazhdan set. Examples of such G include automorphism groups of countable ω-categorical structures, that is, the closed, oligomorphic permutation groups on a countable set. The proof uses work of the second author on the unitary representations of such groups, together with a separation result for infinite permutation...

Free non-archimedean topological groups

Michael Megrelishvili, Menachem Shlossberg (2013)

Commentationes Mathematicae Universitatis Carolinae

We study free topological groups defined over uniform spaces in some subclasses of the class 𝐍𝐀 of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean 𝐍𝐀 groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another...

From double Lie groups to quantum groups

Piotr Stachura (2005)

Fundamenta Mathematicae

It is shown, using geometric methods, that there is a C*-algebraic quantum group related to any double Lie group (also known as a matched pair of Lie groups or a bicrossproduct Lie group). An algebra underlying this quantum group is the algebra of a differential groupoid naturally associated with the double Lie group.

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