Related papers: Formal conjugacy growth in graph products I

We study coherence of graph products and Coxeter groups and obtain many results in this direction.

We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random…

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish…

In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu, Hermiller, Holt and Rees. We focus primarily on virtual graph products in the form of a semi-direct product. First, we…

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…

We show that the class of $\mathcal{C}$-hereditarily conjugacy separable groups is closed under taking arbitrary graph products whenever the class $\mathcal{C}$ is an extension closed variety of finite groups. As a consequence we show that…

Given a finite group $G$, denote by $\Gamma(G)$ the simple undirected graph whose vertices are the distinct sizes of noncentral conjugacy classes of $G$, and set two vertices of $\Gamma(G)$ to be adjacent if and only if they are not coprime…

We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics…

Given two regular graphs with consistent rotation maps, we produce a constructive method for a consistent rotation map on their Cartesian product. This method will be given as a simple set of rules of addition and table look ups. We assume…

Monotone grid classes of permutations have proven very effective in helping to determine structural and enumerative properties of classical permutation pattern classes. Associated with grid class $\mathrm{Grid}(M)$ is a graph, $G(M)$, known…

We describe an effective version of the conjugacy problem and study it for wreath products and free solvable groups. The problem involves estimating the length of short conjugators between two elements of the group, a notion which leads to…

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider…

The objective of this paper is to detect which combinatorial properties of a regular graph can completely determine the geodesic growth of the right-angled Coxeter or Artin group this graph defines, and to provide the first examples of…

We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.

We introduce and study, for a process P delivering edges on the Cartesian product of the vertex sets of a given set of graphs, the P-product of these graphs, thereby generalizing many types of product graph. Analogous to the notion of a…

Let $G$ be a finite group and $a\in G$. Let $a^G=\{g^{-1}ag\mid g\in G\}$ be the conjugacy class of $a$ in $G$. Assume that $a^G$ and $b^G$ are conjugacy classes of $G$ with the property that ${\bf C}_G(a)={\bf C}_G(b)$. Then $a^G b^G$ is a…

We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which…

A new general formula for the number of conjugacy classes of subgroups of given index in a finitely generated group is obtained.