Related papers: The Power Word Problem in Graph Products

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ be its quotient group, e.g. a right-angled Artin or Coxeter group. Given a system of word equations over $M(A,I)$ with recognizable constraints with input size…

In algebraic terms, the insertion of $n$-powers in words may be modelled at the language level by considering the pseudovariety of ordered monoids defined by the inequality $1\le x^n$. We compare this pseudovariety with several other…

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…

Let $P(G)$ denotes the set of sizes of fibers of non-trivial commutators of the commutator word map. Here, we prove that $|P(G)|=1$, for any finite group $G$ of nilpotency class $3$ with exactlly two conjugacy class sizes. We also show that…

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which so-called benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups…

The factor graph of an instance of a symmetric constraint satisfaction problem on n Boolean variables and m constraints (CSPs such as k-SAT, k-AND, k-LIN) is a bipartite graph describing which variables appear in which constraints. The…

The aim of the present note is to construct invariants of the Artin braid group valued in $G_{N}^{2}$, and further study of groups related to $G_{n}^{3}$. In the groups $G_{n}^{2}$, the word problem is solved; these groups are much simpler…

Consider a compact connected Lie group $G$ and the corresponding Lie algebra $\cal L$. Let $\{X_1,...,X_m\}$ be a set of generators for the Lie algebra $\cal L$. We prove that $G$ is uniformly finitely generated by $\{X_1,...,X_m\}$. This…

We construct an automaton group with a PSPACE-complete word problem, proving a conjecture due to Steinberg. Additionally, the constructed group has a provably more difficult, namely EXPSPACE-complete, compressed word problem and acts over a…

A word is called a reset word for a deterministic finite automaton if it maps all the states of the automaton to a unique state. Deciding about the existence of a reset word of a given maximum length for a given automaton is known to be an…

The power graph $\mathcal{P}(G)$ is the simple undirected graph with group elements as a vertex set and two elements are adjacent if one of them is a power of the other. The order supergraph $\mathcal{S}(G)$ of the power graph…

Baumslag-Solitar groups were introduced in 1962 by Baumslag and Solitar as examples for finitely presented non-Hopfian two-generator groups. Since then, they served as examples for a wide range of purposes. As Baumslag-Solitar groups are…

Let $\omega_0, \omega_1,\ldots, \omega_n$ be a full set of outcomes (letters, symbols) and let positive $p_i$, $i=0,\ldots,n$, be their probabilities ($\sum_{i=0}^n p_i=1$). Let us treat $\omega_0$ as a stop symbol; it can occur in…

A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every…

The unitary Cayley graph of $\mathbb{Z} /n \mathbb{Z}$, denoted $X_{\mathbb{Z} / n \mathbb{Z}}$, has vertices $0,1, \dots, n-1$ with $x$ adjacent to $y$ if $x-y$ is relatively prime to $n$. We present results on the tightness of the known…

We investigate partial Equality and Word Problems for finitely generated groups. After introducing Upper Banach (UB) density on free groups, we prove that solvability of the Equality Problem on squares of UB-generic sets implies solvability…

We show that the uniform Constraint Satisfaction Problem (CSP) parameterized by the size of the solution is in W[1] (the problem is W[1]-hard and it is easy to place it in W[3]). Given a single "free" element of the domain, denoted by $0$,…

This paper provides a bridge between two active areas of research, the spectrum (set of element orders) and the power graph of a finite group. The order sequence of a finite group $G$ is the list of orders of elements of the group, arranged…

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of…

The enhanced power graph $\mathcal{P}_e(G)$ of a group $G$ is a graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number and…