Related papers: On discrete homology of a free pro-$p$-group

We prove that for a free noncyclic group $F$, $H_2(\hat F_\mathbb Q, \mathbb Q)$ is an uncountable $\mathbb Q$-vector space. Here $\hat F_\mathbb Q$ is the $\mathbb Q$-completion of $F$. This answers a problem of A.K. Bousfield for the case…

We prove that the second homology group of the completion of an infinite dimensional free Lie algebra is uncountable.

In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic…

We address the following question. For which finitely generated pro-$p$ groups the comparison map $\phi^2:H_{cont}^{2}(P,\F_p) \to H_{disc}{2}(P,\F_p)$ is an isomorphism? We prove that if $P$ is not finitely presented then $\phi^2$ is not…

For a non-cyclic free group $F$, the second homology of its pronilpotent completion $H_2(\widehat F)$ is not a cotorsion group.

We show that the first homology group of a locally connected compact metric space is either uncountable or is finitely generated. This is related to Shelah's well-known result which shows that the fundamental group of such a space satisfies…

It is proved that any countable topological vector space over a finite field $\mathbb F_p$ or, equivalently, any countable Abelian topological group of prime exponent has a closed discrete basis.

A second countable virtually free pro-p group all of whose torsion elements have finite centralizer is the free pro-p product of finite p-groups and a free pro-p factor.

Letting $G=F/R$ be a finitely-presented group, Hopf's formula expresses the second integral homology of $G$ in terms of $F$ and $R$. Expanding on previous work, we explain how to find generators of $H_2(G;\mathbb{F}_p)$. The context of the…

Let $p$ be an odd prime, and let $n\in \N$ be an integer. We show that the $n^{\text{th}}$ mod-$p$ cohomology of a solvable saturable pro-$p$ group is isomorphic to the $n^{\text{th}}$ mod-$p$ cohomology of its associated $\Z_p$-Lie algebra…

We study the commensurators of free groups and free pro-$p$ groups, as well as certain subgroups of these. We prove that the commensurator $Comm(F)$ of a non-abelian free group of finite rank $F$ is not virtually simple, answering a…

This paper is devoted to the computation of the space $H_b^2(\Gamma,H;\mathbb{R})$, where $\Gamma$ is a free group of finite rank $n\geq 2$ and $H$ is a subgroup of finite rank. More precisely we prove that $H$ has infinite index in…

We prove that the group G=Hom(P,Z) of all homomorphisms from the Baer-Specker group P to the group Z of integer numbers endowed with the topology of pointwise convergence contains no infinite compact subsets. We deduce from this fact that…

We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$…

We show that the torsion-free rank of $H_i(M, \mathbb{Z}_p)$ has finite upper bound for $i \leq m$, where $M$ runs through the pro-$p$ subgroups of finite index in a pro-$p$ group $G$ that is (nilpotent of class $c$)-by-abelian such that $…

We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F…

In this note we introduce pro-Hall $R$-groups as inverse limits of Hall $R$-groups and show that for the binomial closure $S^{bin}$ of any ring $S$ discriminated by $\mathbb{Z}_p$, the free pro-Hall $S^{bin}$-group $\mathbb{F}(A,S^{bin})$…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

We classify pointed $p^3$-dimensional Hopf algebras $H$ over any algebraically closed field $k$ of prime characteristic $p>0$. In particular, we focus on the cases when the group $G(H)$ of group-like elements is of order $p$ or $p^2$, that…

We examine second bounded cohomology and mod p homology in finite index subgroups of 1-relator groups and groups with a presentation of deficiency at least one. We use this to determine exactly which 1-relator groups are boundedly…