Related papers: Normalising graphs of groups

Let $\Gamma$ be a finite graph together with a group $G_v$ at each vertex $v$. The graph product $G(\Gamma)$ is obtained from the free product of all $G_v$ by factoring out by the normal subgroup generated by $\{g^{-1}h^{-1}gh; g\in G_v,…

Given a finite group $G$, the generating graph $\Gamma(G)$ of $G$ has as vertices the non-identity elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. Let $G$ be a 2-generated…

These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…

Starting from context-free inverse graphs, we introduce a new class of groups and study their structural properties. We establish closure properties, show that their co-word problems are context-free, analyze torsion elements, and realize…

In this note we prove that finitely generated virtually free groups are stable with respect to a normalized $p$-Schatten norm for $1\leq p < \infty$. In particular, this implies that virtually free groups are Hilbert-Schmidt stable.

The main result of this paper is that, if $\Gamma$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $\Gamma$, then either $\Gamma$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or…

To any graph with external half-edges and internal masses, we associate canonical integrals which depend non-trivially on particle masses and momenta, and are always finite. They are generalised Feynman integrals which satisfy graphical…

It is known that, if removing some $n$ edges from a graph $\Gamma$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $|E(K)|\cdot n$ edges, which form a…

A finitely generated group $\Gamma$ is called strongly scale-invariant if there exists an injective endomorphism $\varphi: \Gamma \to \Gamma$ with the image $\varphi(\Gamma)$ of finite index in $\Gamma$ and the subgroup $\displaystyle…

We study countable graphs that -- up to isomorphism and with probability one -- arise from a random process, in a similar fashion as the Rado graph. Unlike in the classical case, we do not require that probabilities assigned to pairs of…

Let G be a finite graph with the non-k-order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of…

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…

We count the finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$. More precisely: each such subgroup $H$ can be represented by its Stallings graph $\Gamma(H)$, we consider the number of vertices of $\Gamma(H)$ to…

We consider a number of examples of groups together with an infinite conjugation invariant generating set, including: the free group with the generating set of all separable elements; surface groups with the generating set of all…

We study a class of finite groups $G$ which behave similarly to elementary abelian $p$-groups with $p$ prime, that is, there exists a subgroup $N$ such that all elements of $G\setminus N$ are conjugate or inverse-conjugate under $\Aut(G)$.…

We describe the groups that have the same holomorph as a finite perfect group. Our results are complete for centerless groups. When the center is non-trivial, some questions remain open. The peculiarities of the general case are illustrated…

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and $\Gamma_C$ the Cayley graph of $G$ based on $C \cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded…

We prove that the isomorphism problem for finitely generated fully residually free groups is decidable. We also show that each finitely generated fully residually free group G has a decomposition that is invariant under automorphisms of G,…

Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include…

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for…