International Journal of Algebra and Computation

We consider quotients of the free product G of groups A and B, amalgamated along a group C, which is malnormal in A and in B. We concentrate on quotients H of G by the normal closure of a single relator which is a power of a word. We show that a part of the results which are true for the corresponding one relator free products hold true for H. We reduce the problems to the free product case and use small cancellation theory.

The follow automaton of a regular expression, recently introduced by Ilie and Yu, is a quotient of the position automaton. The aim of this paper is to present an efficient computation of this quotient, based on specific properties of the structure of the expression.

We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson–Higman group G_3,1.

In this work we give pairs of generators (x, y) for the alternating groups A_n, 5 ≤ n ≤ 19, such that they determine the minimal genus of a Riemann surface on which A_n acts as the automorphism group. Using these results we prove that A₁₅ is the unique of these groups that is an H*-group, i.e., the groups achieving the upper bound of the order of an automorphism group acting on non-orientable unbordered surfaces.

Let G be a group hyperbolic relative to a collection of subgroups {H_λ, λ ∈ Λ}. We say that a subgroup Q ≤ G is hyperbolically embedded into G, if G is hyperbolic relative to {H_λ, λ ∈ Λ} ∪ {Q}. In this paper we obtain a characterization of hyperbolically embedded subgroups. In particular, we show that if an element g ∈ G has infinite order and is not conjugate to an element of some H_λ, λ ∈ Λ, then the (unique) maximal elementary subgroup containing g is hyperbolically embedded into G. This allows us to prove that if G is boundedly generated, then G is elementary or H_λ = G for some λ ∈ Λ.

We show that a number of natural membership problems for classes associated with finite semigroups are computationally difficult. In particular, we construct a 55-element semigroup S such that the finite membership problem for the variety of semigroups generated by S interprets the graph 3-colorability problem.

We characterize the structure of linear semigroups satisfying certain global and local nilpotence conditions and deduce various Engel-type results. For example, using a form of Zel'manov's solution of the restricted Burnside problem we are able to show that a finitely generated residually finite group is nilpotent if and only if it satisfies a certain 4-generator property of semigroups we call WMN. Methods of linear semigroups then allow us to prove that a linear semigroup is Mal'cev nilpotent precisely when it satisfies WMN. As an application, we show that a finitely generated associative algebra is nilpotent when viewed as a Lie algebra if and only if its adjoint semigroup is WMN.

In 1997, following the works of Klaus W. Wagner on ω-regular sets, Olivier Carton and Dominique Perrin introduced the notions of chains and superchains for ω-semigroups. There is a clear correspondence between the algebraic representation of each of these operations and the automata-theoretical one. Unfortunately, chains and superchains do not suffice to describe the whole Wagner hierarchy. We introduce a third notion that completes the task undertaken by these two authors.

Let I denote the commutator ideal in the free associative algebra on m variables over an arbitrary field. In this article we prove there are exactly m! finite Gröbner bases for I, and uncountably many infinite Gröbner bases for I with respect to total division orderings. In addition, for m = 3 we give a complete description of its universal Gröbner basis.

The author previously introduced groups BV and as braided versions of groups introduced by Richard J. Thompson, and as having connections to braided tensor categories. Here we show that BV and are finitely presented by exhibiting finite presentations for them. We show that these groups are "Artinifications" of the older Thompson groups V and by showing that presentations for V and can be obtained from presentations for BV and by adding relations that trivialize the squares of the generators of the latter groups.