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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-4,
author = {David M. Evans and Todor Tsankov},
title = {Free actions of free groups on countable structures and property (T)},
journal = {Fundamenta Mathematicae},
volume = {233},
year = {2016},
pages = {49-63},
zbl = {1337.22004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-4}
}
David M. Evans; Todor Tsankov. Free actions of free groups on countable structures and property (T). Fundamenta Mathematicae, Tome 233 (2016) pp. 49-63. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm232-1-4/