The conjugacy problem for HNN groups and the word problem for commutative semigroups

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in \(\mathsf {NP}\). We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove \(\mathsf {NP}\)-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class \(\mathsf {qAC}^k\) (for some \(k \in \mathbb {N}\)) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.

A graph product is the fundamental group of a graph of groups Amongst the simplest examples are HNN groups and free products with amalgamation.The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical.For graph products over infinite graphs these conditions are insufficient: a further condition ensures that graph products over infinite graphs of bounded path length have solvable conjugacy problem.These results generalise the known ones for HNN groups and free products with amalgamation.

The word problem of a finitely generated commutative semigroup is shown to be context-free if and only if this semigroup does not contain the free commutative semigroup on two generators as a subsemigroup, answering a question of Brough et al. An analogous characterization is established for monoids as well.

Uniform decision problems for automatic semigroups

Our purpose is to show the equivalence of the conjugacy problem for certain HNN extensions of the infinite cyclic groups and the reachability problem for the class of self-dual vector addition systems. In addition, we extend an endomorphism theorem of the author’s to a homomorphism theorem and indicate a problem related to the isomorphism problem for a class of HNN groups.

SynopsisThe small cancellation theory over free products with amalgamation and HNN groups is extended to groups acting on trees in which the action with inversions is possible. This will include the case of tree products of groups and treed-HNN groups.

This paper shows that the equivalence problems for commutative semigroups and symmetric vector addition systems are decidable in space cNlogN for some fixed constant c, solving an open question by Cardoza, Lipton, Mayr, and Meyer. From the exponential-space completeness of the word problems, it follows that our upper bound is nearly optimal.

The isomorphism problem for commutative semigroups

Introduction

We propose a way of associating to each finitely generated monoid or semigroup a formal language, called its loop problem. In the case of a group, the loop problem is essentially the same as the word problem in the sense of combinatorial group theory. Like the word problem for groups, the loop problem is regular if and only if the monoid is finite. We also study the case in which the loop problem is context-free, showing that a celebrated group-theoretic result of Muller and Schupp extends to describe completely simple semigroups with context-free loop problems. We consider right cancellative monoids, establishing connections between the loop problem and the structural theory of these semigroups by showing that the syntactic monoid of the loop problem is the inverse hull of the monoid.

The three fundamental decision problems posed by Max Dehn in 1912 are the word problem, the conjugacy problem and the isomorphism problem. Let G be a group and let u and v be elements of G. The word problem asks for an algorithm for deciding whether u = v. The conjugacy problem asks about the existence of an element g ∈ G which conjugates u to v, i.e., v = g −1 ug. A solution of the conjugacy problem clearly contains a solution of the word problem. The isomorphism problem asks whether two group presentations define isomorphic groups. The word and conjugacy problems received much attention in the literature. For a summary of results concerning the conjugacy problem until 1987 see [13] and references therein.

An algorithm is constructed which solves the word problem for special semigroups with overlapping of defining words less than 1/3.Bibliography: 4 titles. Figures: 10.

Amenability of Schreier graphs and strongly generic algorithms for the conjugacy problem

The conjugacy problem asks whether two words over generators of a fixed group G are conjugated, i.e., it is the problem to decide on input words x, y whether there exists z such that $$zx z^{-1} =y$$zxz-1=y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the conjugacy problem for two prominent groups: the Baumslag---Solitar group $$\mathbf{{BS}}_{1,2}$$BS1,2 and the Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 (also known as Baumslag---Gersten group). The conjugacy problem in $${\mathbf{{BS}}}_{1,2}$$BS1,2 is complete for the circuit class $$\mathsf {TC}^0$$TC0. To the best of our knowledge $${\mathbf{{BS}}}_{1,2}$$BS1,2 is the first natural infinite non-commutative group where such a precise and low complexity is shown. The Baumslag group $${\mathbf{{G}}}_{1,2}$$G1,2 is an HNN-extension of $${\mathbf{{BS}}}_{1,2}$$BS1,2. Hence, decidability of the conjugacy problem in $$\mathbf{{G}}_{1,2}$$G1,2 outside the so-called black hole follows from Borovik et al. (Int J Algebra Comput 17(5/6):963---997, 2007). Decidability everywhere is due to Beese. Moreover, he showed exponential time for the set of elements which cannot be conjugated into $$\mathbf{{BS}}_{1,2}$$BS1,2 (Beese 2012). Here we improve Beese's result in two directions by showing that the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 can be solved in polynomial time in a strongly generic setting. This means that essentially for all inputs, conjugacy in $${\mathbf{{G}}}_{1,2}$$G1,2 can be decided efficiently. In contrast, we show that under a plausible assumption the average case complexity of the same problem is non-elementary. Moreover, we provide a lower bound for the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2 by reducing the divisibility problem in power circuits to the conjugacy problem in $${\mathbf{{G}}}_{1,2}$$G1,2. The complexity of the divisibility problem in power circuits is an open and interesting problem in integer arithmetic. We conjecture that it cannot be solved in elementary time because we can show that it cannot be solved in elementary time by calculating modulo values in power circuits.

The decision problems in the title are solved using the solutions of the word and conjugacy problems obtained by Lyndon and Schupp, respectively, for certain classes of groups.