We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N). Some other cases when G/N has virtually finite cohomological dimension are considere...

Let $S$ be a semitopological semigroup. The $wap-$ compactification of semigroup S, is a compact semitopological semigroup with certain universal properties relative to the original semigroup, and the $Lmc-$ compactification of semigroup $S$ is a universal semigroup compactification of $S$, which are denoted by $S^{wap}$ and $S^{Lmc}$ respectively. In this paper, an internal construction of ...

in the present paper we give a partially negative answer to a conjecture of ghahramani, runde and willis. we also discuss the derivation problem for both foundation semigroup algebras and clifford semigroup algebras. in particular, we prove that if s is a topological clifford semigroup for which es is finite, then h1(m(s),m(s))={0}.

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finite products, subalgebras and quotients. In this paper Reiterman’s theorem is generalised to finite Eilenberg-Moore algebras for a monad T on a variety D of (ordered) algebras: ...

Profinite semigroups may be described shortly as projective limits of finite semigroups. They come about naturally by studying pseudovarieties of finite semigroups which in turn serve as a classifying tool for rational languages. Of particular relevance are relatively free profinite semigroups which for pseudovarieties play the role of free algebras in the theory of varieties. Combinatorial pro...

In the present work, we give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Also we show that right-angled Artin groups are residually torsion-free nilpotent. Moreover, we investigate the profinite topology of F2 × F2 and of the group L in [18], which are the only obstructions for the subgroup separab...

It is well known [Hoc69, Joy71] that profinite T0-spaces are exactly the spectral spaces. We generalize this result to the category of all topological spaces by showing that the following conditions are equivalent: (1) (X,τ) is a profinite topological space. (2) The T0-reflection of (X,τ) is a profinite T0-space. (3) (X,τ) is a quasi spectral space (in the sense of [BMM08]). (4) (X,τ) admits a ...

This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In §2 we collect some questions that motivated the lectures and this article, and in §3 discu...

We provide elementary proofs of the Nielsen-Schreier Theorem and the Kurosh Subgroup Theorem via wreath products. Our proofs are diagrammatic in nature and work simultaneously in the abstract and profinite categories. A new proof that open subgroups of quasifree profinite groups are quasifree is also given.