An example of a non-Borel locally-connected finite-dimensional topological group

Answering a question posed by S.Maillot in MathOverFlow, for every $n\in\mathbb N$ we construct a locally connected subgroup $G\subset\mathbb R^{n+1}$ of dimension $dim(G)=n$, which is not locally compact.Comment: 2 page

On closed embeddings of free topological algebras

Let $\mathcal K$ be a complete quasivariety of completely regular universal topological algebras of continuous signature $\mathcal E$ (which means that $\mathcal K$ is closed under taking subalgebras, Cartesian products, and includes all completely regular topological $\mathcal E$-algebras algebraically isomorphic to members of $\mathcal K$). For a topological space $X$ by $F(X)$ we denote the free universal $\mathcal E$-algebra over $X$ in the class $\mathcal K$. Using some extension properties of the Hartman-Mycielski construction we prove that for a closed subspace $X$ of a metrizable (more generally, stratifiable) space $Y$ the induced homomorphism $F(X)\to F(Y)$ between the respective free universal algebras is a closed topological embedding. This generalizes one result of V.Uspenskii concerning embeddings of free topological groups.Comment: 3 page

The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $\Gamma\subset[0,\infty)$, we recognize the topological structure of the system (2^X,\D_{\le\gamma}(X))_{\alpha\in\Gamma}, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\le\gamma}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\le\gamma$.Comment: 12 page

The coarse classification of countable abelian groups

We prove that two countable locally finite-by-abelian groups G,H endowed with proper left-invariant metrics are coarsely equivalent if and only if their asymptotic dimensions coincide and the groups are either both finitely-generated or both are infinitely generated. On the other hand, we show that each countable group G that coarsely embeds into a countable abelian group is locally nilpotent-by-finite. Moreover, the group G is locally abelian-by-finite if and only if G is undistorted in the sense that G can be written as the union of countably many finitely generated subgroups G_n such that each G_n is undistorted in G_{n+1} (which means that the identity inclusion from G_n to G_{n+1} is a quasi-isometric embedding with respect to word metrics).Comment: 25 pages. Longer version with new results about FCC groups, locally finite-by-abelian groups, locally nilpotent-by-finite groups