Related papers: The finitely generated intersection property in fu…

The $G$-graph $\Gamma(G,S)$ is a graph from the group $G$ generated by $S\subseteq G$, where the vertices are the right cosets of the cyclic subgroups $\langle s \rangle, s\in S$ with $k$-edges between two distinct cosets if there is an…

The main result in this paper is the failure of the finitely generated intersection property (FGIP) of ascending HNN extensions of non-cyclic finite rank free groups. This class of group consists of free-by-cyclic groups and properly…

We establish conditions under which the fundamental group of a graph of finite $p$-groups is necessarily residually $p$-finite. The technique of proof is independent of previously established results of this type, and the result is also…

For a finite connected graph $\mathcal{E}$ with set of edges $E$, a finite $E$-generated group $G$ is constructed such that the set of relations $p=1$ satisfied by $G$ (with $p$ a word over $E\cup E^{-1}$) is closed under deletion of…

In this article we generalize the theory of subgroup graphs of subgroups of free groups to finite index subgroups $H$ of finitely generated groups $G$. We study and prove various properties of $H$ in relation to its subgroup graph…

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…

This paper studies intersections of principal blocks of a finite group with respect to different primes. We first define the block graph of a finite group $G$, whose vertices are the prime divisors of $|G|$ and there is an edge between two…

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental…

Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by $\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only if the…

We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z…

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…

Suppose $G$ is a finitely presented group that is hyperbolic relative to ${\bf P}$ a finite collection of 1-ended finitely generated proper subgroups of $G$. If $G$ and the ${\bf P}$ are 1-ended and the boundary $\partial (G,{\bf P})$ has…

We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called…

A group $G$ is said to be intersection-saturated if for every strictly positive integer $n$ and every map $c\colon \mathcal{P}(\{1,\dots, n\})\setminus \emptyset \rightarrow \{0,1\}$, one can find subgroups $H_1,\dots, H_n\leq G$ such that…

Let $G$ be a relatively hyperbolic group that admits a decomposition into a finite graph of relatively hyperbolic groups structure with quasi-isometrically (qi) embedded condition. We prove that the set of conjugates of all the vertex and…

In an earlier work, finite groups whose power graphs are minimally edge connected have been classified. In this article, first we obtain a necessary and sufficient condition for an arbitrary graph to be minimally edge connected.…

We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups…

We determine the structure of the finite groups with the property that every cyclic subgroup is the intersection of maximal subgroups, comparing this property with the one where all proper subgroups are intersections of maximal subgroups.

Let $G$ be a group. The intersection graph $\Gamma(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two…

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…