Related papers: Evaluation problems for the Thompson group and the…

We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin--Dehornoy braided Thompson group $bV$ has an infinite-dimensional space of quasimorphisms and thus…

The co-word problem of a group G generated by a set X is defined as the set of words in X which do not represent 1 in G. We introduce a new method to decide if a permutation group has context-free co-word problem. We use this method to…

The Thompson-Higman groups G_{k,i} have a natural generalization to monoids M_{k,i}, and inverse monoids Inv_{k,i}. We study some structural features of M_{k,i} and Inv_{k,i} and investigate the computational complexity of decision…

We consider generalisations of Thompson's group $V$, denoted by $V_r(\Sigma)$, which also include the groups of Higman, Stein and Brin. It was shown by the authors in [20] that under some mild conditions these groups and centralisers of…

Let $\mathrm{WP}_G$ denote the word problem in a finitely generated group $G$. We consider the complexity of $\mathrm{WP}_G$ with respect to standard deterministic Turing machines. Let $\mathrm{DTIME}_k(t(n))$ be the complexity class of…

We prove that the compressed word problem and the compressed simultaneous conjugacy problem are solvable in polynomial time in hyperbolic groups. In such problems, group elements are input as words defined by straight line programs defined…

A monoid is called special if it admits a presentation in which all defining relations are of the form $w = 1$. Every group is special, but not every monoid is special. In this article, we describe the language-theoretic properties of the…

We study the monoid generalization M_{k,1} of the Thompson-Higman groups, and we characterize the R- and the L-preorder of M_{k,1}. Although M_{k,1} has only one non-zero J-class and k-1 non-zero D-classes, the R- and the L-preorder are…

We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using,…

We construct an extension $E(A,G)$ of a given group $G$ by infinite non-Archimedean words over an discretely ordered abelian group like $Z^n$. This yields an effective and uniform method to study various groups that "behave like $G$". We…

We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be…

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive…

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating…

The word problem is an old and central problem in (computational) group theory. It is well-known that the word problem is undecidable in general, but decidable for specific types of presentations. Consistent polycyclic presentations are an…

Let $F_n$ be a free group of rank $n$. In this paper we discuss three algorithmic problems related to automorphisms of $F_2$. A word $u$ of $F_n$ is called positive if $u$ does not have negative exponents. A word $u$ in $F_n$ is called…

Group classification of a class of Benjamin-Bona-Mahony (BBM) equations with time dependent coefficients is carried out. Two equivalent lists of equations possessing Lie symmetry extensions are presented: up to point equivalence within the…

This paper continues the functional approach to the P-versus-NP problem, begun in [1]. Here we focus on the monoid RM_2^P of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We…

We show that the compressed word problem in a finitely-generated fully residually free group (F -group) is decidable in polynomial time, and use the result to show that the word problem in the automorphism group of such a group is decidable…

We prove that the generalised word problem of a finitely generated subgroup of a finitely generated virtually free group is context-free, that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of…

We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show…