Related papers: On pro-${\it p}$-groups with a single defining rel…

We show that if 2^{aleph_0} Cohen reals are added to the universe, then for every reduced non-free torsion-free abelian group A of cardinality less than the continuum, there is a prime p so that Ext_p(A, Z) not= 0. In particular if it is…

We introduce the condition of a profinite group being semi-free, which is more general than being free and more restrictive than being quasi-free. In particular, every projective semi-free profinite group is free. We prove that the usual…

For a subgroup of a free product of finite groups, we obtain necessary conditions (on its Kurosh decomposition) to be verbally closed.

A subset $X$ of a finite group $G$ is said to be prime-power-independent if each element in $X$ has prime power order and there is no proper subset $Y$ of $X$ with $\langle Y, \Phi(G)\rangle = \langle X, \Phi(G)\rangle$, where $\Phi(G)$ is…

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of…

A group is known as `large' if some finite index subgroup admits a surjective homomorphism onto a non-abelian free group. In this paper, we give a necessary and sufficient condition for a finitely presented group to be large, in terms of…

Given a finitely generated group $\Gamma$ and $g\in\Gamma$, we prove sufficient conditions in terms of various growth/decay functions for freeness of the action of $g$ on the Furstenberg boundary of $\Gamma$. In this context, we also give a…

We discuss whether finiteness properties of a profinite group $G$ can be deduced from the coefficients of the probabilistic zeta function $P_G(s)$. In particular we prove that if $P_G(s)$ is rational and all but finitely many non abelian…

It is proved that generalized free product of two finite p-groups is a conjugacy p-separable group if and only if it is residually finite p-groups. This result is then applied to establish some sufficient conditions for conjugacy…

We give a counterexample to a conjecture by Miasnikov, Ventura and Weil, stating that an extension of free groups is algebraic if and only if the corresponding morphism of their core graphs is onto, for every basis of the ambient group. In…

We study analogues of classical Hilbert transforms as fourier multipliers on free groups. We prove their complete boundedness on non commutative $L^p$ spaces associated with the free group von Neumann algebras for all $1<p<\infty$. This…

In this article, for a polyadic group(G,f),derived from group G by automorphism G and element b, we give a necessary and sufficient condition in terms of the group, the automorphism G, and the element b, in order that the polyadic group…

Let G be a finitely generated infinite pro-p group acting on a pro-p tree such that the restriction of the action to some open subgroup is free. Then we prove that G splits as a pro-p amalgamated product or as a pro-p HNN-extension over an…

We prove the pro-$p$ version of the Karras, Pietrowski, Solitar, Cohen and Scott result stating that a virtually free group acts on a tree with finite vertex stabilizers. If a virtually free pro-$p$ group $G$ has finite centralizes of all…

Let $G=\{e^{tA}:t\in\mathbb{R}\}$ be a closed one-parameter subgroup of the general linear group of matrices of order $n$ acting on $\mathbb{R}^{n}$ by matrix-vector multiplications. We assume that all eigenvalues of $A$ are rationally…

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the…

The celebrated Stallings' decomposition theorem states that the splitting of a finite index subgroup $H$ of a finitely generated group $G$ as an amalgamated free product or an HNN-extension over a finite group implies the same for $G$. We…

We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.

In this note we consider a $p$-isometrisability property of discrete groups. If $p=2$ this property is equivalent to unitarisability. We prove that any group containing a non-abelian free subgroup is not $p$-isometrisable for any $p\in (1,…

We prove the Identity Theorem for pro-$p$-groups with a single defining relation giving a positive feedback to a question of Serre on the structure of relation modules. A construction of "conjurings" indicates finality of our result in a…