Related papers: When does a right-angled Artin group split over $\…

Let $\Gamma$ be the random bipartite graph, a countable graph with two infinite sides, edges randomly distributed between the sides, but no edges within a side. In this paper, we investigate the reducts of $\Gamma$ that preserve sides. We…

Given a 2-generated finite group $G$, the non-generating graph of $G$ has as vertices the elements of $G$ and two vertices are adjacent if and only if they are distinct and do not generate $G$. We consider the graph $\Sigma(G)$ obtained…

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n \geq 4$, every balanced bipartite graph on $2n$…

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V(\Gamma)$ admits a nontrivial $G$-invariant partition ${\cal B}$ such…

We study the outer automorphism group of a right-angled Artin group A_G in the case where the defining graph G is connected and triangle-free. We give an algebraic description of Out(A_G) in terms of maximal join subgraphs in G and prove…

We study the outer automorphism group of a right-angled Artin group $A_\Gamma$ with finite defining graph $\Gamma$. We construct a subnormal series for $Out(A_\Gamma)$ such that each consecutive quotient is either finite, free-abelian,…

We associate a graph $\mathcal{C}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\backslash Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G | < x,y> \text{is cyclic for all} y\in G\}$ is called…

In this paper we investigate properties of the Artin monoid Cayley graph. This is the Cayley graph of an Artin group $A_\Gamma$ with respect to the (infinite) generating set given by the associated Artin monoid $A^+_\Gamma$. In a previous…

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this…

In this paper, we give a characterization of unicyclic graphs with diameter at most 4 which are A-vertex magic. Moreover, let G be a bicyclic graph of diameter 3, then G is group vertex magic if and only if G = M11(0, 0).

There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows…

There is an established bijection between finite-index subgroups Gamma of Gamma(2) and bipartite graphs on surfaces, or, equivalently, certain triples of permutations. We utilize this relationship to study both congruence and noncongruence…

We characterize, in terms of the defining graph, when a twisted right-angled Artin group (a group whose only relations among pairs of generators are either commuting or Klein-bottle type relations) is left-orderable.

We prove three results about the graph product $G=\G(\Gamma;G_v, v \in V(\Gamma))$ of groups $G_v$ over a graph $\Gamma$. The first result generalises a result of Servatius, Droms and Servatius, proved by them for right-angled Artin groups;…

For $n\in \mathbb{N}$, a group is called $n$-coherent if every subgroup of type $\mathsf{F}_n$ is of type $\mathsf{F}_{n+1}$. For $n\ge 1$, we observe that graphs of groups with $n$-coherent vertex groups and virtually poly-cyclic edge…

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…

We characterize which groups splitting as finite graphs of free groups with cyclic edge groups are residually finite. Such a group $G$ is residually finite if and only if all its Baumslag-Solitar subgroups are residually finite. From a…

Given a group $G$ and an automorphism $\varphi$ of $G$, two elements $x, y \in G$ are said to be $\varphi$-conjugate if $x = g y \varphi(g)^{-1}$ for some $g \in G$. The number of equivalence classes is the Reidemeister number $R(\varphi)$…

We are motivated by the question that for which class of right-angled Artin groups (RAAG's), the quasi-isometry classification coincides with commensurability classification. This is previously known for RAAG's with finite outer…

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these…