Journal
FUNDAMENTA MATHEMATICAE
Volume 232, Issue 1, Pages 49-63Publisher
POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN
DOI: 10.4064/fm232-1-4
Keywords
oligomorphic groups; property (T); omega-categoricity; unitary representations; Roelcke precompact
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Funding
- ANR grant GrupoLoco [ANR-11-JS01-008]
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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 omega-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.
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