Related papers: Nilpotent groups with balanced presentations

Let G be a group and N be the class of nilpotent groups. A subset A of G is said to be nonnilpotent if for any two distinct elements a and b in A, ha, bi 62 N. If, for any other nonnilpotent subset B in G, |A| ? |B|, then A is said to be a…

It is proved that for any prime $p$ a finitely generated nilpotent group is conjugacy separable in the class of finite $p$-groups if and only if the torsion subgroup of it is a finite $p$-group and the quotient group by the torsion subgroup…

We show that a finitely generated abelian group $G$ of torsion-free rank $n\geq 1$ admits a $n+r$ dimensional model for the classifying space with isotropy in the family of subgroups of torsion-free rank less than or equal to $r\geq 0$.

Let $G$ be a group. The orbits of the natural action of Aut$(G)$ on $G$ are called ``automorphism orbits'' of $G$, and the number of automorphism orbits of $G$ is denoted by $\omega(G)$. We prove that if $G$ is a soluble group with finite…

We give a new obstruction to translation-like actions on nilpotent groups. Suppose we are given two finitely generated torsion free nilpotent groups with the same degree of polynomial growth, but non-isomorphic Carnot completions…

We examine the first non-vanishing higher homotopy group, $\pi_p$, of the complement of a hypersolvable, non--supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\Z\pi_1$. We give a…

Suppose that G is a nontrivial torsion-free group and w is a word in the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\} such that the word w' obtained from w by erasing all letters belonging to G is not a proper power in the free group…

In this paper we consider the palindromic width of free nilpotent groups. In particular, we prove that the palindromic width of a finitely generated free nilpotent group is finite. We also prove that the palindromic width of a free…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not…

It is known that an abelian group $A$ and a $2$-cocycle $c:A \times A \to C$ yield a group ${\mathscr{H}}(A,C,c)$ which we call a Heisenberg group. This group, a central extension of $A$, is the archetype of a class~$2$ nilpotent group. In…

Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + \kappa(\omega)$ exists, then $\mbox{TFAG}$ is $a \Delta^1_2$-complete; in particular, this is consistent with…

In this note we prove the claim given in the title. A group G is noncommutatively slender if each map from the fundamental group of the Hawaiian Earring to G factors through projection to a canonical free subgroup. Graham Higman, in his…

We prove that the mapping class group $\mathcal{M}(N_g)$ of a closed nonorientable surface of genus $g$ different than 4 is generated by three torsion elements. Moreover, for every even integer $k\ge 12$ and $g$ of the form $g=pk+2q(k-1)$…

Let $FH$ be a supersolvable Frobenius group with kernel $F$ and complement $H$. Suppose that a finite group $G$ admits $FH$ as a group of automorphisms in such a manner that $C_G(F)=1$ and $C_{G}(H)$ is nilpotent of class $c$. We show that…

We present an explicit structure for the Baer invariant of a free $n$th nilpotent group (the $n$th nilpotent product of infinite cyclic groups, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*...\st{n}*\textbf{Z}$) with respect to the variety ${\cal…

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

We give a complete classification of Einstein Lorentzian 3-nilpotent simply connected Lie groups with 1-dimensional nondegenerate center.

We show that every box space of a virtually nilpotent group has asymptotic dimension equal to the Hirsch length of that group.

We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means…

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in…