We prove that the quasivariety of groups generated by finite and locally indicable groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit-Thompson theorem on the solvability of finite groups of odd order, Kostrikin-Zelmanov results on the restricted Burnside problem and applies techninal details of a recent construction of weakly finitely presented periodic groups.
Systems of equations and their solution sets are studied in polyadic groups. We prove that a polyadic group $(G, f)=\mathrm{der}_{θ, b}(G, \cdot)$ is equational noetherian, if and only if the ordinary group $(G, \cdot)$ is equational noetherian. The structure of coordinate polyadic group of algebraic sets in equational noetherian polyadic groups are also determined.
We attach with every finite, involutive, nondegenerate set-theoretic solution of the Yang--Baxter equation a finite group that plays for the associated structure group the role that a finite Coxeter group plays for the associated Artin--Tits group.
In this article, we study the property of being equationally Artinian in groups. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form $G[t]$ by a normal subgroup which is a finite union of radicals, is again equationally Artnian. This provides a large class of examples of equationally Artinian groups.
We prove that, for every integer $n \ge 2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, i.e., every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.
The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions of hyperbolic groups is solvable.
We give the following characterization of sofic (weakly sofic) groups: a group $G$ is sofic (weakly sofic) if and only if any system of equations solvable in any alternating group (any finite group) is solvable over $G$.
We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.
We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations solvable in all alternating groups is solvable over $G$. The last characterization allows to express soficity of an existentially closed group by $\forall\exists$-sentences. Keywords: sofic groups, approximations, equations over groups.
I prove the "folklore" result that the functional equation for a Lie group homomorphism can be solved by solving the corresponding differential equation.
We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm for solving equations and inequations in hyperbolic groups (maybe with torsion). Additionnally, we can deal with quasi-isometrically embeddable rational constraints.
In this note, we show that the satisfiability of equations and inequations with recognisable constraints is decidable in groups that are virtually direct products of finitely many hyperbolic groups.
We provide an algorithm which, for a given quadratic equation in the Grigorchuk group determines if it has a solution. As a corollary to our approach, we prove that the group has a finite commutator width.
We provide polynomial upper bounds on the size of a shortest solution for quadratic equations in a free group. A similar bound is given for parametric solutions in the description of solutions sets of quadratic equations in a free group.
The aim of this paper is to compare and contrast the class of residually finite groups with the class of equationally Noetherian groups - groups over which every system of coefficient-free equations is equivalent to a finite subsystem. It is easy to construct groups that are residually finite but not equationally Noetherian (e.g. the direct sum of all finite groups) or vice versa (e.g. the additive group $(\mathbb{Q},+)$ of the rationals). However, no explicit examples that are finitely generated seem to appear in the literature. In this paper, we show that among finitely generated groups, the…
The Levin conjecture was proposed by Levin in 1962 which conjectures the solvability of any group equation with coefficients in a torsion free group. The Levin conjecture is recently shown to hold for group equations of length seven by weight test and curvature distribution. These methods are applied on a group equation of length nine to explore the validity of Levin conjecture. It is found that the Levin conjecture has an affirmative answer for this equation modulo some exceptional cases.
We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.
We prove that in a free group the length of the value of each variable in a minimal solution of a standard quadratic equation is bounded by $2s$ for orientable equation and by $12s^4$ for non-orientable equation, where $s$ is the sum of the lengths of the coefficients
We show that the class of groups where EDT0L languages can be used to describe solution sets to systems of equations is closed under direct products, wreath products with finite groups, and passing to finite index subgroups. We also add the class of groups that contain a direct product of hyperbolic groups as a finite index subgroup to the list of groups where solutions to systems of equations can be expressed as an EDT0L language. This includes dihedral Artin groups. We also show that the systems of equations with rational constraints in virtually abelian groups have EDT0L solutions, and the …
In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of equations over a free ${\bf Q}$-group has a solution, and if it does, finds a solution. This statement can be generalized for ${\bf Q}$-completions of torsion-free hyperbolic groups. Our proof significantly uses the techniques of word hyperbolic groups and the results of E.Rips and Z.Sela on the solvability of systems of equations in hyperbolic groups.
Here we give a complete group classification of the general case of linear systems of three second-order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second-order ordinary differential equations. In addition the complete group classification of a system of three linear second-order ordinary differential equations is done. Four cases of linear systems of equations are obtained.
We investigate the solution sets to equations in the solvable Baumslag-Solitar groups $BS(1,k)$, $k\geq2$, and show that these sets are represented by EDT0L languages in some cases. In particular, we prove that the multiplication table of such a group forms an EDT0L language with respect to a specific natural normal form for group elements.
The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than …
We give a new criterion for solvability of group equations, providing proofs of various generalizations of the Kervaire-Laudenbach conjecture for Connes-embeddable groups.
It is shown that, for any pair of cardinals with infinite sum, there exist a group and an equation over this group such that the first cardinal is the number of solutions to this equation and the second cardinal is the number of non-solutions to this equation. A countable torsion-free non-topologizable group is constructed.
The present note shows that $\mathcal{Q}$-groups in [H. Heineken and F.G. Russo, Groups described by element numbers, Forum Math. 27 (2015), 1961--1977] are solvable groups (not necessarily nilpotent) for which the equation $T_G(r,s)=0$ is satisfied.
The notion of a symmetrically factorizable Lie group is introduced. It is shown that each symmetrically factorizable Lie group is related to a set-theoretical solution of the pentagon equation. Each simple Lie group (after a certain Abelian extension) is symmetrically factorizable.
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiability problem in finite groups by showing that it is in P for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups of Fitting length two. In this work, we present the first lower bounds for the equation satisfiability problem in finite solvable groups: under the assumption of the exponential time hypothesis, we show that it cannot be in P for any group of Fitting len…
We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations …
The study of the complexity of the equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (2002) where they showed that this problem is in polynomial time for nilpotent groups while it is NP-complete for non-solvable groups. Since then, several results have appeared showing that the problem can be solved in polynomial time in certain solvable groups $G$ having a nilpotent normal subgroup $H$ with nilpotent factor $G/H$. This paper shows that such normal subgroup must exist in each finite group with equation satisfiability solvable in polynomial time, unless…
In this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$-evolution equations for higher order operators, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, de Giorgi and Spagnolo. In particular, we describe an interesting loss of regularity phenomenon depen…
Let $Γ$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $Γ$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $Γ$ is free.
The survey provides an overview of the work done in the last 10 years to characterise solutions to equations in groups in terms of formal languages. We begin with the work of Ciobanu, Diekert and Elder, who showed that solutions to systems of equations in free groups in terms of reduced words are expressible as EDT0L languages. We provide a sketch of their algorithm, and describe how the free group results extend to hyperbolic groups. The characterisation of solutions as EDT0L languages is very robust, and many group constructions preserve this, as shown by Levine. The most recent progress in …
This paper represents notes of the mini-courses given by the authors at the GCGTA conference in Dortmund (2007), Ottawa-Saint Sauveur conference (2007), Escola d'Algebra in Rio de Janeiro (2008) and Alagna (Italy, 2008) conference on equations in groups. We explain here the Elimination process for solving equations in a free group which has Makanin-Razborov process as a prototype. We also explain how we use this process to obtain the structure theorem for finitely generated fully residually free groups and many other results.
In this paper we begin the systematic study of group equations with abelian predicates in the main classes of groups where solving equations is possible. We extend the line of work on word equations with length constraints, and more generally, on extensions of the existential theory of semigroups, to the world of groups. We use interpretability by equations to establish model-theoretic and algebraic conditions which are sufficient to get undecidability. We apply our results to (non-abelian) right-angled Artin groups, and show that the problem of solving equations with abelian predicates is und…
Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests a fractional Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
We give a brief survey of recent results on word maps on simple groups and polynomial maps on simple associative and Lie algebras. Our focus is on parallelism between these theories, allowing one to state many new open problems and giving new ways for solving older ones.
This paper contains a survey of recent developments in investigation of word equations in simple matrix groups and polynomial equations in simple (associative and Lie) matrix algebras along with some new results on the image of word maps on algebraic groups defined over special fields: complex, real, p-adic (or close to such), or finite.
In this paper we study the problem of classification of indecomposable solutions of the Yang-Baxter equation. Using a scheme proposed by Bachiller, Cedó, and Jespers, and recent advances in the classification of braces we classify all indecomposable solutions with some particular permutation groups. We do this for all groups of size $pq$, all abelian groups of size $p^2q$ and all dihedral groups of size $p^2q$.
Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion equation is used as an example to illustrate the effectiveness of the Lie group method.
We continue our previous investigation of the Zelisko group of a matrix over Bézout domains. The explicit form of elements of this group over homomorphic image of Bézout domain of stable rank 1.5 is described.
