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

David M. Evans ¹   ; Todor Tsankov ²

¹ Department of Mathematics Imperial College London London SW7 2AZ, UK
² Université Paris 7 UFR de Mathématiques, Case 7012 75205 Paris Cedex 13, France

Fundamenta Mathematicae, Tome 232 (2016) no. 1, pp. 49-63

Résumé

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 $G$.

Détail

DOI : 10.4064/fm232-1-4

Keywords: non archimedean roelcke precompact polish group has kazhdans property moreover has smallest subgroup finite index has finite kazhdan set examples include automorphism groups countable omega categorical structures closed oligomorphic permutation groups countable set proof uses work second author unitary representations groups together separation result infinite permutation groups latter allows construction non abelian subgroup acting freely infinite transitive permutation representations