Related papers: Concise presentations of direct products

Let $G$ be the right-angled Artin group associated to the flag complex $\Sigma$ and let $\pi:G\to\Z$ be its canonical height function. We investigate the presentation theory of the groups $\Gamma_n=\pi^{-1}(n\Z)$ and construct an algorithm…

It is a classical result that the direct product AxB of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not…

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…

Let N be a normal subgroup of a finite group G. We prove that under certain (unavoidable) conditions the subgroup [N,G] is a product of commutators [N,y] (with prescribed values of y from a given set Y) of length bounded by a function of…

Let G be a Lie group, $g = Lie(G)$ - its Lie algebra, $g*$ - the dual vector space and $\widehat G$ - the set of equivalence classes of unitary irreducible representations of $G$. The orbit method [1] establishes a correspondence between…

So far, it has been proven that if $G$ is an abelian group , then the diameter of $G^n$ with respect to any generating set is $O(n)$; and if $G$ is nilpotent, symmetric or dihedral, then there exists a generating set of minimum size, for…

We describe generators and defining relations for the commutator subgroup of topological full groups of minimal subshifts. We show that the word problem in a topological full group is solvable if and only if the language of the underlying…

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is…

A result by Bridson, Howie, Miller, and Short states that if $S$ is a finitely presented subgroup of the direct product of free groups, then $S$ is virtually a nilpotent extension of a direct product of free groups. Moreover, if $S$ is a…

Given any finitely presented group G we find a triangular algebra such that has two presentations, one with fundamental group G and another with trivial group. Thus proving that given a collection G1,...,Gn of finitely presented groups…

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $\beta_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then…

Hard to summarize concisely; here are the high points. The first two statements below are ring-theoretic; in these R is a nontrivial ring, R^\omega, and \bigoplus_\omega R are the direct product, respectively direct sum, of countably many…

The Cayley table representation of a group uses $\mathcal{O}(n^2)$ words for a group of order $n$ and answers multiplication queries in time $\mathcal{O}(1)$. It is interesting to ask if there is a $o(n^2)$ space representation of groups…

Let $G$ be a finite solvable group. Then $G$ always has a useful presentation, which we call a "long presentation". Using a "long presentation" of $G$, we present an inductive method of constructing the irreducible representations of $G$…

Let $G$ be a finitely generated group. We prove that the $n$-fold tensor product $G^{\otimes n}$ is finite (resp. polycyclic) if and only $G$ is finite (resp. polycyclic). Further, assuming that $G$ is finitely presented, we show that…

We construct examples of finitely generated decidable group presentations that satisfy certain combinations of solvability for the word problem, solvability for the bounded word problem, and computablity for the Dehn function. We prove that…

Let $G$ be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of $G$ with a given infinitesimal character and a…

Hougthon's groups H_n is a family of groups where each H_n consists of `translations at infinity' on n rays of discrete points emanating from the origin on the plane. Brown shows H_n has type FP_n-1 but not FP_n by constructing infinite…

Let $Q_m$ be the HNN extension of $\Z/m \times \Z/m$ where the stable letter conjugates the first factor to the second. We explore small presentations of the groups $\Gamma_{m,n}=Q_m \ast Q_n$. We show that for certain choices of (m,n), for…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (a)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…