Related papers: Common transversals for coset spaces of compact gr…

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which…

In a previous article by the author and P. Wesolek, it was shown that a compactly generated locally compact group $G$ admits a finite normal series $(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no…

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_\sigma$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to…

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We show that, under mild conditions on a…

L\'evai and Pyber proposed the following as a conjecture: Let $G$ be a profinite group such that the set of solutions of the equation $x^n=1$ has positive Haar measure. Then $G$ has an open subgroup $H$ and an element $t$ such that all…

We explore transversals of finite index subgroups of finitely generated groups. We show that when $H$ is a subgroup of a rank $n$ group $G$ and $H$ has index at least $n$ in $G$ then we can construct a left transversal for $H$ which…

Let $G$ be a topological group and let $K,L\subseteq G$ be closed subgroups, with $K\subseteq L$. We prove that if $L$ is a locally compact pro-Lie group, then the map $q:G/K\to G/L$ is a fibration. As an application of this, we obtain two…

Let $G$ be an abelian Polish group. We show that there is a strongly Haar meager set in $G$ without any $F_{\sigma}$ Haar meager hull (and that this still remains true if we replace $F_{\sigma}$ by any other class of the Borel hierarchy).…

It is known that for every second countable locally compact group G, there exists a proper G-invariant metric which induces the topology of the group. This is no longer true for coset spaces G/H viewed as G-spaces. We study necessary and…

It is introduced a certain approach for equipment of an arbitrary set of the cardinality of the continuum by structures of Polish groups and two-sided (left or right) invariant Haar measures. By using this approach we answer positively…

For a Polish group $G$, let $E(G)$ be the right coset equivalence relation $G^\omega/c(G)$, where $c(G)$ is the group of all convergent sequences in $G$. We prove a Rigid theorem on locally compact TSI Polish groups admitting open identity…

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual…

For a representation of a connected compact Lie group G in a finite dimensional real vector space U and a subspace V of U, invariant under a maximal torus of G, we obtain a sufficient condition for V to meet all G-orbits in U, which is also…

Let $X$ be the countable product of Abelian locally compact Polish groups and $A,B\subset X$ be two Borel sets, which are not Haar-null in $X$. We prove that the sum-set $A+B:=\{a+b:a\in A,\;\;b\in B\}$ is Haar-open in the sense that for…

A topological space is almost locally compact if it contains a dense locally compact subspace. We generalize a result from \cite{Ma}, showing that isomorphism on Borel classes of almost locally compact Polish metric structures is always…

Let $H$ be a closed normal subgroup of a compact Lie group $G$ such that $G/H$ is connected. This paper provides a necessary and sufficient condition for every complex representation of $H$ to be extendible to $G$, and also for every…

A set $E$ in a Banach space $X$ is compactivorous if for every compact set $K$ in $X$ there is a nonempty, (relatively) open subset of $K$ which can be translated into $E$. In a separable Banach space, this is a sufficient condition which…

If $K$ is a closed subgroup of a compact group $G$, the probability that randomly chosen pair of elements from $K$ and $G$ commute is denoted by $Pr(K,G)$. Say that a subgroup $K\leq G$ is $\epsilon$-central in $G$ if $Pr(\langle g…

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$,…

We extend some results of Carderi and Le Ma\^itre on full groups in the probability context to the infinite measure one: there exists at most one Polish group topology (refining the weak topology and coarser than the uniform topology) on an…