Related papers: The completeness of free Boolean topological group…

We say that a group $G$ has Bergman's property (the property of universality of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have that $G=X^k$ for some natural number $k.$ The property is named after George…

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild…

We prove that a self-similar free abelian group has finite rank. We apply the result to self-similar wreath products of abelian groups $G=BwrX$. We show that if $X$ is torsion-free, then $B$ is torsion of finite exponent. Furthemore, we…

In this note we show that a $p$-complete nilpotent space $X$ has a $p$-complete suspension spectrum if and only if its homotopy groups $\pi_*X$ are bounded $p$-torsion. In contrast, if $\pi_*X$ is not all bounded $p$-torsion, we locate…

We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid…

A completeness conjecture is advanced concerning the free small-colimit completion P(A) of a (possibly large) category A. The conjecture is based on the existence of a small generating-cogenerating set of objects in A. We sketch how the…

The object of this expository work is to try to unveil the topological/geometric intuition behind the theory of free groups and their automorphism and outer automorphism groups. The method we follow is to focus on a series of problems in…

In this paper we construct a new "pro-p-complete" topological Kac-Moody group and compare it to various known topological Kac-Moody groups. We come across this group by investigating the process of completion of groups with BN-pairs. We…

If A is an abelian variety over a number field K, and L is a (possibly infinite) extension of K generated by torsion points of A, then the quotient of A(L) by its torsion subgroup is a free abelian group.

We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor…

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,…

We construct a Taylor-Wiles system using a family of $p$-adically completed modular Jacobians over suitable Hecke algebras and prove that certain $p$-adically completed Mordell-Weil groups of these Jacobians is free of finite rank over a…

Let X be a separable metric space and let \beta be the strict topology on the space of bounded continuous functions on X, which has the space of \tau-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonne\'{e} type…

Let G be a finite Chevalley group and B a Borel subgroup. Then the interval [B,G] in L(G) is Boolean. We prove, using Zsigmondy's theorem, that for any element P in the open interval (B,G), its lattice-complement P^c is not a…

In this article we prove that the set of torsion-free groups acting by isometries on a hyperbolic metric space whose entropy is bounded above and with a compact quotient is finite. The number of such groups can be estimated in terms of the…

A Boolean algebra carries a strictly positive exhaustive submeasure if and only if it has a sequential topology that is uniformly Frechet.

In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of…

We construct finitely generated torsion-free solvable groups $G$ that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of $G$ are virtually abelian. In particular all finitely generated…

In this paper we focus on the set-open topologies on the group $\mathcal{H}(X)$ of all self-homeomorphisms of a topological space $X$ which yield continuity of both the group operations, product and inverse function. As a consequence, we…

Given a topological group $G$ that can be embedded as a topological subgroup into some topological vector space (over the field of reals) we say that $G$ has invariant linear span if all linear spans of $G$ under arbitrary embeddings into…