Related papers: Isoperimetric functions for graph products

The Gruenberg--Kegel graph (or the prime graph) $\Gamma(G)$ of a finite group $G$ is defined as follows. The vertex set of $\Gamma(G)$ is the set of all prime divisors of the order of $G$. Two distinct primes $r$ and $s$ regarded as…

There are a variety of ways to associate directed or undirected graphs to a group. It may be interesting to investigate the relations between the structure of these graphs and characterizing certain properties of the group in terms of some…

We prove that every recurrent graph $G$ quasi-isometric to $\mathbb{R}$ admits an essentially unique Lipschitz harmonic function $h$. If $G$ is vertex-transitive, then the action of $Aut(G)$ preserves $\partial h$ up to a sign, a fact that…

Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpi\'{n}ski product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$,…

We show that if $G$ is a group and $G$ has a graph-product decomposition with finitely-generated abelian vertex groups, then $G$ has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

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;…

Let $G$ be a finite insoluble group with soluble radical $ R(G)$. The solubility graph $\Gamma_{\rm S}(G)$ of $G$ is a simple graph whose vertices are the elements of $G\setminus R(G) $ and two distinct vertices $x$ and $y$ are adjacent if…

To each finite frame $\varphi$ in an inner product space $\mathcal{H}$ we associate a simple graph $G(\varphi)$, called {\it frame graph}, with the vectors of the frame as vertices and there is an edge between vertices $f$ and $g$ provided…

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In…

In this paper, we investigate certain graphs defined on groups, with a focus on infinite groups. The graphs discussed are the power graph, the enhanced power graph, and the commuting graph whose vertex set is a group $G$. The power graph is…

Let $G_\Gamma$ be a graph product over a finite simplicial graph $\Gamma$, and let $K_\Gamma$ denote the kernel of the canonical homomorphism from $G_\Gamma$ to the direct product of its vertex groups. It is known that, up to isomorphism,…

In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are identical…

Given a finite group $G,$ we denote by $\Delta(G)$ the graph whose vertices are the proper subgroups of $G$ and in which two vertices $H$ and $K$ are joined by an edge if and only if $G=\langle H,K\rangle.$ We prove that if there exists a…

Given a finite connected simple graph $\Gamma$, and a subgroup $G$ of its automorphism group, a general method for finding all finite abelian regular coverings of $\Gamma$ that admit a lift of each element of $G$ is developed. As an…

A graph $G$ is called universal for a family of graphs $\mathcal{F}$ if it contains every element $F \in \mathcal{F}$ as a subgraph. Let $\mathcal{F}(n,2)$ be the family of all graphs with maximum degree $2$. Ferber, Kronenberg, and Luh…

This is the first of a sequence of papers devoted to studying the link between the complexity of the Word Problem for a finitely generated recursively presented group $G$ and the isoperimetric functions of the finitely presented groups in…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

We show that every effectively closed action of a finitely generated group $G$ on a closed subset of $\{0,1\}^{\mathbb{N}}$ can be obtained as a topological factor of the $G$-subaction of a $(G \times H_1 \times H_2)$-subshift of finite…