Projectively universal countable metrizable groups

More Polish full groups

Computable topological abelian groups

We replace measure with category in an argument of G. W. Mackey to characterize closed subgroups H of a totally nonmeager, 2nd countable topological group G in terms of the quotient Borel structure G/H.As a corollary, we obtain an improved version of a theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish groups.In [9], G. W. Mackey gave an argument using Haar measures to prove the following: Assume G is locally compact topological group.If H is a subgroup such that the space G/H, formed by giving the set of (left) H-cosets the quotient Borel structure, is countably separated, then H is closed in G.We will show that this result can be obtained by an analogous argument using the theory of Baire category.The category version has a wider application and shows that the above statement remains true under the weaker assumption that G is totally nonmeager.In particular, it holds whenever G is topologically complete (in the sense of Cech).As a corollary, we will prove that a well-known theorem of C. Ryll-Nardzewski on the Borel measurability of orbits in continuous actions by Polish (separable, completely metrizable) groups holds for Borel actions as well.We will also show that a recent result of R. Vaught on decompositions of invariant analytic sets has a similar extension to Borel actions.Each step in our main argument corresponds to an analogous step in Mackey's proof.The reader may find it interesting to compare our development with the proof of Theorem 7.2 in Mackey [9].Recall (cf.[2, IX,) that a topological group G which is not meager (first category) is a Baire space (i.e.no nonempty open subset of G is meager in G).G is totally nonmeager if no nonempty closed subset of G is meager in itself.Every topologically complete space, in particular every locally compact space and every completely metrizable space, is totally nonmeager (cf.[7, III, 8]).B E G is strictly Baire (or has the restricted Baire property)

Some aspects of dimension theory for topological groups

We show that all groups of a distinguished class of «large» topological groups, that of Roelcke precompact Polish groups, have Kazhdan’s Property (T). This answers a question of Tsankov and generalizes previous results by Bekka (for the infinite-dimensional unitary group) and by Evans and Tsankov (for oligomorphic groups). Further examples include the group \({{\,\mathrm{Aut}\,}}(\mu )\) of measure-preserving transformations of the unit interval and the group \({{\,\mathrm{Aut}\,}}^*(\mu )\) of non-singular transformations of the unit interval.More precisely, we prove that the smallest cocompact normal subgroup \(G^\circ \) of any given non-compact Roelcke precompact Polish group G has a free subgroup \(F\le G^\circ \) of rank two with the following property: every unitary representation of \(G^\circ \) without invariant unit vectors restricts to a multiple of the left-regular representation of F. The proof is model-theoretic and does not rely on results of classification of unitary representations. Its main ingredient is the construction, for any \(\aleph _0\)-categorical metric structure, of an action of a free group on a system of elementary substructures with suitable independence conditions.

Graev metric groups and Polishable subgroups

Necessary and sufficient conditions are given for a Polish topological group to be ‘almost free’. It is deduced that the existence of one free subgroup of a Polish group can lead to the existence of many free subgroups of maximal rank. Applications are given to permutation groups, profinite groups, Lie groups and unitary groups.

Let $M=I$ or $M=\mathbb{S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop{\rm Diff}_+^{k+AC}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$-th derivative is absolutely continuous, admits a natural Polish group topology which refines the subspace topology inherited from $\mathop{\rm Diff}_+^k(M)$. By contrast, the group $\mathop{\rm Diff}_+^{1+BV}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.

Bounded normal generation and invariant automatic continuity

Totally Lindelöf and totally ω-narrow paratopological groups

A mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

Spaces of maps into topological group with the Whitney topology

We investigate properties of the clopen type semigroup of an action of a countable group on a compact, 0 -dimensional, Hausdorff space X . We discuss some characterizations of dynamical comparison (most of which were already known in the metrizable case) in this setting and prove that for a Cantor minimal action \alpha of an amenable group the topological full group of \alpha admits a dense, locally finite subgroup iff the corresponding clopen type semigroup is unperforated. We also discuss some properties of clopen type semigroups of the Stone–Čech compactifications and universal minimal flows of countable groups, and derive some consequences on generic properties in the space of minimal actions of a given countable group on the Cantor space.

Precompact groups and convergence

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.