The Solubility Graph Associated With a Finite Group

Some properties of various graphs associated with finite groups

In this paper we have investigated some properties of the power graph and commuting graph associated with a finite group, using their tree-numbers. Among other things, it has been shown that the simple group $L_2(7)$ can be characterized through the tree-number of its power graph. Moreover, the classification of groups with power-free decomposition is presented. Finally, we have obtained an explicit formula concerning the tree-number of commuting graphs associated with the Suzuki simple groups.

Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud' in their book "Characters of finite groups, Part $1$", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character coedgree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classif…

The Reidemeister spectrum of 2-step nilpotent groups determined by graphs

In this paper we study the Reidemeister spectrum of 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group in terms of the Reidemeister spectra of groups associated to smaller graphs. We illustrate our methods for several families of graphs, including all the groups associated to a graph with at most four vertices. We also apply our results in the context of topological fixed point theory for nilmanifolds.

Acylindrical hyperbolicity of Artin-Tits groups associated to triangle-free graphs and cones over square-free bipartite graphs

It is conjectured that the central quotient of every irreducible Artin group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin groups associated to triangle-free graphs and Artin groups of large type associated to cones over square-free bipartite graphs. In fact, we treat Artin groups that are known to be CAT(0) groups by a result of Brady and McCammond.

Cartan subalgebras in von Neumann algebras associated with graph product groups

In this short note we classify the Cartan subalgebras in all von Neumann algebras associated with graph product groups and their free ergodic measure preserving actions on probability spaces.

Characterizing finite nilpotent groups associated with a graph theoretic equality

The power graph of a group is the simple graph whose vertices are the group elements and two vertices are adjacent whenever one of them is a positive power of the other. We characterize the finite nilpotent groups whose power graphs have equal vertex connectivity and minimum degree.

Acylindrical hyperbolicity of Artin groups associated with graphs that are not cones

Charney and Morris-Wright showed acylindrical hyperbolicity of Artin groups of infinite type associated with graphs that are not joins, by studying clique-cube complexes and actions on them. In this paper, by developing their study and formulating some additional discussion, we demonstrate that acylindrical hyperbolicity holds for more general Artin groups. Indeed, we are able to treat Artin groups of infinite type associated with graphs that are not cones.

一类与自相似群相关的中间增长Schreier图

On a family of Schreier graphs of intermediate growth associated with a self-similar group

对于任意的无穷序列$ω=x_1,x_2,...$，利用$x_i\in\{0,1\}$，我们构造了一个无穷4-正则图$X_ω$ .这些图正是空间$\{0,1\}^{\infty}$上某个自相似群作用的Schreier图。我们解决了这些图的同构和局部同构问题，并确定了它们的自同构群。最后证明了所有图$X_ω$都具有中间增长性。

Reconstructible graphs, simplicial flag complexes of homology manifolds and associated right-angled Coxeter groups

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $Γ$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $Γ$ is reconstructible.