Related papers: On paratopological groups

A topological group $G$ is called an $M_\omega$-group if it admits a countable cover $\K$ by closed metrizable subspaces of $G$ such that a subset $U$ of $G$ is open in $G$ if and only if $U\cap K$ is open in $K$ for every $K\in\K$. It is…

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that…

In this note it is proved that the class of paratopologies is simple and that under the assumption that the measurable cardinals form a proper class, the class of hypotopologies is not simple. Moreover, an example is given of a Hausdorff…

In \cite{Kramer11} Kramer proves for a large class of semisimple Lie groups that they admit just one locally compact $\sigma$-compact Hausdorff topology compatible with the group operations. We present two different methods of generalising…

We generalize a classical result of Andrianov on the decomposition of Hecke polynomials. Let $\mathfrak{F}$ be a non-archimedean local fied. For every connected reductive group $\mathbf{G}$, we give a criterion for when a polynomial with…

A Hausdorff topological group $(G,\tau)$ is called an $s$-group and $\tau$ is called an $s$-topology if there is a set $S$ of sequences in $G$ such that $\tau$ is the finest Hausdorff group topology on $G$ in which every sequence of $S$…

Let G be an exceptional algebraic group defined over an algebraically closed field k of characteristic p>0 and let H be a subgroup of G. Then following Serre we say H is G-completely reducible or G-cr if, whenever H is contained in a…

There are many deep results on the structure of REGULAR probability measures $P(G)$ on compact/locally compact, Hausdorff topological groups G. See, for instance, the classic monographs by KR Parthasarathy, Ulf Grenander, A.Mukherjea and…

A semitopological group $G$ is called {\it an $n$-semitopological group}, if for any $g\in G$ with $e\not\in\overline{\{g\}}$ there is a neighborhood $W$ of $e$ such that $g\not\in W^{n}$, where $n\in\mathbb{N}$. The class of…

In this work we have considered the complexity of the different structures as topological group on Z. We collect some new results, as well as some known results on the group of the integers in order to present: -A family of $2^\cont$…

Let Gamma be an S-arithmetic subgroup of a solvable algebraic group G over an algebraic number field F, such that the finite set S contains at least one place that is nonarchimedean. We construct a certain group H, such that if L is any…

When does a topological group $G$ have a Hausdorff compactification $bG$ with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A.V. Arhangel'ski\v{\i} and C. Liu's. Let $G$ be a non-locally…

We show that for any finitely generated subgroup $H$ of a limit group $L$ there exists a finite-index subgroup $K$ containing $H$, such that $K$ is a subgroup of a group obtained from $H$ by a series of extensions of centralizers and free…

Building on results of Medvedev, we construct a $\mathsf{ZFC}$ example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of $\mathbb{Z}^\omega$ of size $\mathfrak{b}$ that is a…

We prove that if $H$ is a topological group such that all closed subgroups of $H$ are separable, then the product $G\times H$ has the same property for every separable compact group $G$. Let $c$ be the cardinality of the continuum. Assuming…

A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $…

We establish that finitely generated non-abelian direct products $G$ of free pro-$p$ groups have full Hausdorff spectrum with respect to the lower $p$-series $\mathcal{L}$. This complements similar results with respect to other standard…

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1,…

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It…

It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology…