Graev metric groups and Polishable subgroups

Long colimits of topological groups II: Free groups and vector spaces

More Polish full groups

Computable topological abelian groups

If the group G=AB is the product of two abelian subgroups A and B, then G is metabelian by a well-known result of Itô [8], so that the commutator subgroup G' of G is abelian. In the following we are concerned with the following condition:There exists a normal subgroupwhich is contained in A or B.Recently, Holt and Howlett in [7] have given an example of a countably infinite p-group G = AB, which is the product of two elementary abelian subgroups A and B with Core(A) = Core (B) = 1, so that in this group (*) does not hold. Also, Sysak in [13] gives an example of a product G = AB of two free abelian subgroups A and B with Core(A)=Core(B)=l.

Most of the results in this paper concern relationships between sequential properties of a pointed topological space (X, p) and sequential properties of the Graev free topological group on X. In particular, it is shown that the free group over a sequential ${k_\omega }$-space is sequential, and that a nondiscrete sequential free group has sequential order equal to ${\omega _1}$ (the first uncountable ordinal). The free topological group on a space X which includes a convergent sequence contains a closed subspace homeomorphic to ${S_\omega }$, a previously studied homogeneous, zero-dimensional sequential space. Finally, it is shown that there is no topological group homeomorphic to ${S_\omega }$.

Weakly complete free topological groups

Non-periodic groups without free Abelian subgroups of rank 2 with non-Dedekind norm of Abelian noncyclic subgroups are studied.

The topological fundamental group and free topological groups

The topological properties of q-spaces in free topological groups

Notes on countable tightness of the subspaces of free (Abelian) topological groups

We construct free abelian subgroups of the group U(AΓ) of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group U(AΓ) was studied in [5] by constructing a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bounds agree. In many cases when the principal cubes are not of maximal dimension we show there is an invariant contractible subcomplex of strictly lower dimension.

We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.

Some aspects of dimension theory for topological groups

We prove that there exists a countable metrizable topological group G G such that every countable metrizable group is isomorphic to a quotient of G G . The completion H H of G G is a Polish group such that every Polish group is isomorphic to a quotient of H H .

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.