Related papers: Fixed Price of Groups and Percolation

The aim of this small note is to prove an elementary yet useful properties of finitely presented groups. Let G be a finitely generated group with one end. Fix a (finite) generating set and let $B_n$ be the ball of radius $n$ around $e$. Let…

Let $\Sigma=(\Gamma, \sigma)$ is a signed graph(or sigraph in short), where $\Gamma$ is a underlying graph of $\Sigma$ and $\sigma:E\longrightarrow \{+, -\}$ is a function. Consider $\Gamma=Cay(\mathbb{Z}_{p_{1}}\times…

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…

We show that an infinite finitely generated group G is virtually-Z if and only if every Cayley graph of G contains only finitely many Busemann points in its horofunction boundary. This complements a previous result of the second named…

Given a finite simplicial graph $\Gamma=(V,E)$ with a vertex-labelling $\varphi:V\rightarrow\left\{\text{non-trivial finitely generated groups}\right\}$, the graph product $G_\Gamma$ is the free product of the vertex groups $\varphi(v)$…

Let $\Gamma$ be a finite graph and let $A(\Gamma)$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A(\Gamma)$ is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. Here we…

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…

Let S=Sym(\Omega) be the group of all permutations of a countably infinite set \Omega, and for subgroups G_1, G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that < G_1\cup U > = < G_2\cup U >. It is…

We prove that any finite subgroup $G \subset \Gamma_{{\boldsymbol{s}}}$ of the cluster modular group has fixed points in the cluster manifolds $\mathcal{A}_{\boldsymbol{s}}(\mathbb{R}_{>0})$ and…

In this article, we show that given any integer $l\geq 2$, every closed curve $\gamma$ on the bouquet of $n$-circles $\Gamma$, admits a lift to a finite $l$-sheeted normal covering of $\Gamma$. Equivalently, identifying the free group $F_n$…

We prove that the invariably generating graph of a finite group can have an arbitrarily large number of connected components with at least two vertices.

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy…

It is shown, for a given graph group $G$, that the fixed point subgroup Fix$\,\varphi$ is finitely generated for every endomorphism $\varphi$ of $G$ if and only if $G$ is a free product of free abelian groups. The same conditions hold for…

We show that there exists a connected graph G with subexponential volume growth such that critical percolation on the product of G with the line has infinitely many infinite clusters. We also give some conditions under which this cannot…

We have defined and established a theory of cofinite connectedness of a cofinite graph. Many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. We have seen that if $G$ is a cofinite group and…

Let $\mathcal G$ be the Cayley graph of a finitely generated, infinite group $\Gamma$. We show that $\Gamma$ has the Haagerup property if and only if for every $\alpha<1$, there is a $\Gamma$-invariant bond percolation $\mathbb P$ on…

We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to…

We prove that every finitely generated residually finite group $G$ can be embedded in a finitely generated branch group $\Gamma$ such that two elements in $G$ are conjugate in $G$ if and only if they are conjugate in $\Gamma$. As an…

The Gruenberg-Kegel graph $\Gamma(G)$ associated with a finite group $G$ has as vertices the prime divisors of $|G|$, with an edge from $p$ to $q$ if and only if $G$ contains an element of order $pq$. This graph has been the subject of much…

We extend some of the fundamental results about percolation on unimodular nonamenable graphs to nonunimodular graphs. We show that they cannot have infinitely many infinite clusters at critical Bernoulli percolation. In the case of heavy…