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Compactifications of ℕ and Polishable subgroups of $S_{∞}$

100%

Tsankov T.

Fundamenta Mathematicae

2006

tom 189

nr 3

269-284

We study homeomorphism groups of metrizable compactifications of ℕ. All of those groups can be represented as almost zero-dimensional Polishable subgroups of the group $S_{∞}$. As a corollary, we show that all Polish groups are continuous homomorphic images of almost zero-dimensional Polishable subgroups of $S_{∞}$. We prove a sufficient condition for these groups to be one-dimensional and also study their descriptive complexity. In the last section we associate with every Polishable ideal on ℕ a certain Polishable subgroup of $S_{∞}$ which shares its topological dimension and descriptive complexity.

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

63%

Evans D., Tsankov T.

Fundamenta Mathematicae

2016

tom 232

nr 1

49-63

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.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Tsankov T.

1 Evans D.

1 2016

1 2006

Wyniki wyszukiwania

Compactifications of ℕ and Polishable subgroups of $S_{∞}$

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