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Preprint Number 666
666. David M. Evans and Todor Tsankov
Free actions of free groups on countable structures and property (T)
Submission date: 18 December 2013.
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 groups. The latter allows the construction of a non-abelian free subgroup of G acting freely in all infinite transitive permutation representations of G.
Mathematics Subject Classification: Primary 22A25, 20B27, Secondary 03C15
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