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Let $G$ be a connected and simply connected complex semisimple Lie group. For a collection of homogeneous $G$-spaces $G/Q$, we construct a finite atlas ${\mathcal{A}}_{\rm BS}(G/Q)$ on $G/Q$, called the Bott-Samelson atlas, and we prove…

In this note, a necessary and sufficient condition for the normalizer of a core-free subgroup $H$ of a finite group $G$ to be normal in $G$ is obtained. Also, a known result of finite groups is obtained through transversal.

We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product…

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its…

Let \(G\) be a non-discrete, locally compact group with Haar measure \(m\). We prove that there exists a compact set \(K \subset G\) with \(m(K)=0\) such that \(KK^{-1}\) contains a neighborhood of the identity. Moreover, such a set may be…

First we give a new proof of Goto's theorem for Lie algebras of compact semisimple Lie groups using Coxeter transformations. Namely, every $x$ in $L = \operatorname{Lie}(G)$ can be written as $x =[a, b]$ for some $a$, $b$ in $L$. By using…

A right Engel sink of an element $g$ of a group $G$ is a subset containing all sufficiently long commutators $[...[[g,x],x],\dots ,x]$. We prove that if $G$ is a compact group in which, for some $k$, every commutator $[...[g_1,g_2],\dots…

In this note, we analyze the classification problem for compact metrizable $G$-ambits for a countable discrete group $G$ from the point of view of descriptive set theory. More precisely, we prove that the topological conjugacy relation on…

We prove a conjecture due to Baumgaertel and Lledo according to which for every compact group G one has Z(G)^ \cong C(G), where the `chain group' C(G) is the free abelian group (written multiplicatively) generated by the set G^ of…

A topological space TO(S) of total orderings on any given set S is introduced and it is shown that TO(S) is compact if S is countable. The set NO(N) of all normal orderings of the nth Weyl algebra W is a closed subspace of TO(N), where N is…

We compute the full holonomy group of compact Lorentzian manifolds with parallel Weyl tensor, which are neither conformally flat nor locally symmetric, for the case where the fundamental group is contained in a distinguished subgroup G of…

The Gromov-Lawson-Rosenberg conjecture for a group G states that a compact spin manifold with fundamental group G admits a metric of positive scalar curvature if and only if a certain topological obstruction vanishes. It is known to be true…

Following Davies, Elekes and Keleti, we study measured sets, i.e. Borel sets $B$ in $\mathbb{R}$ (or in a Polish group) for which there is a translation invariant Borel measure assigning positive and \sigma-finite measure to $B$. We…

Let X be compact abelian group and G its dual (a discrete group). If B is an infinite subset of G, let C_B be the set of all x in X such that <phi(x) : phi \in B> converges to 1. If F is a free filter on G, let D_F be the union of all the…

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related…

We look at the Bohr topology of maximally almost periodic groups (MAP, for short). Among other results, we investigate when a totally bounded abelian group $(G,w)$ is the Bohr reflection of a locally compact abelian group. Necessary and…

Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order; when G is left orderable, this holds if and only if H is convex in G under some left ordering…

The existence of a countably compact group without non-trivial convergent sequences in ZFC alone is a major open problem in topological group theory. We give a ZFC example of a Boolean topological group G without non-trivial convergent…

A discrete subset $S$ of a topologically gyrogroup $G$ is called a {\it suitable set} for $G$ if $S\cup \{1\}$ is closed and the subgyrogroup generated by $S$ is dense in $G$, where $1$ is the identity element of $G$. In this paper, we…

For a subset $A$ of a Polish group $G$, we study the (almost) packing index $\ind_P(A)$ (resp. $\Ind_P(A)$) of $A$, equal to the supremum of cardinalities $|S|$ of subsets $S\subset G$ such that the family of shifts $\{xA\}_{x\in S}$ is…