The aim of this paper is to prove that the Word Problem and the Conjugacy Problem for the structure left skew brace associated with a finite non-degenerate solution of the Yang–Baxter equation are solvable. In order to achieve this result, we need to introduce the concept of (almost) polycyclic left skew brace and to develop a general theory showing that almost polycyclic left skew braces are controlled by their finite homomorphic images. Our results provide us with the first class of infinite solutions of the Yang–Baxter equation on which is possible to work in an algorithmic way the class of almost polycyclic solutions.

Abstract We give a reduction of the conjugacy problem among outer automorphisms of free (and torsion-free hyperbolic) groups to specific algorithmic problems pertaining to mapping tori of polynomially growing automorphisms. We explain how to use this reduction and solve the conjugacy problem for several new classes of outer automorphisms. This proposes a path towards a full solution to the conjugacy problem for ${\textrm{Out}\left (F_{n}\right )}$.

The conjugacy problem and canonical representatives in finitely generated nilpotent groups

In this paper, we investigate the finiteness properties of subgroups of direct products of 2-dimensional coherent right-angled Artin groups. We explore how these properties relate to the structure of the subgroups and the decidability of certain algorithmic problems. More precisely, we show that a finitely presented subgroup S of the direct product of 2-dimensional coherent RAAGs is virtually a nilpotent extension of a direct product. Moreover, if S is of type FP, then S is commensurable to a kernel of a character. We use these results to show that the multiple conjugacy problem and the membership problem are decidable for finitely presented subgroups of direct products of 2-dimensional coherent RAAGs. This work generalizes the results of Bridson, Howie, Miller, and Short for free groups.

This is essentially the only way in which a one-edge extension can be "reducible".We record a specific consequence of this in the following lemma.Lemma 3.4 Suppose that F.H 1 ;

In this paper we provide an alternative solution to a result by Juh\'{a}sz that the twisted conjugacy problem for odd dihedral Artin groups is solvable, that is, groups with presentation $G(m) = \langle a,b \; | \; _{m}(a,b) = {}_{m}(b,a) \rangle$, where $m\geq 3$ is odd, and $_{m}(a,b)$ is the word $abab \dots$ of length $m$, is solvable. Our solution provides an implementable linear time algorithm, by considering an alternative group presentation to that of a torus knot group, and working with geodesic normal forms. An application of this result is that the conjugacy problem is solvable in extensions of odd dihedral Artin groups.Comment: Published in the journal of Groups, Complexity, Cryptology

Abstract We study a family of Thompson-like groups built as rearrangement groups of fractals introduced by Belk and Forrest in 2019, each acting on a Ważewski dendrite. Each of these is a finitely generated group that is dense in the full group of homeomorphisms of the dendrite (studied by Monod and Duchesne in 2019) and has infinite-index finitely generated simple commutator subgroup, with a single possible exception. More properties are discussed, including finite subgroups, the conjugacy problem, invariable generation and existence of free subgroups. We discuss many possible generalisations, among which we find the Airplane rearrangement group $T_A$ . Despite close connections with Thompson’s group F , dendrite rearrangement groups seem to share many features with Thompson’s group V .

In this paper natural necessary and sufficient conditions for quantifier elimination of matrix rings Mn(K) in the language of rings expanded by two unary functions, naming the trace and transposition, are identified. This is used together with invariant theory to prove quantifier elimination when K is an intersection of real closed fields. On the other hand, it is shown that finding a natural definable expansion with quantifier elimination of the theory of Mn(C) is closely related to the infamous simultaneous conjugacy problem in matrix theory. Finally, for various natural structures describing dimension-free matrices it is shown that no such elimination results can hold by establishing undecidability results.

The article considers the solution of the problem of power conjugacy of words in generalized tree structures of Artin groups by geometric methods based on the study of diagrams over this class of groups having a single-layer structure, as previously shown by the authors. Chart transformations are used, including abbreviations introduced by M. Den and V. N. Bezverkhnim.The article is a continuation of the consideration of algorithms for solving problems of combinatorial group theory in generalized tree structures of Artin groups, previously the authors proposed algorithms based on a diagram approach to solve conjugacy problems, generalized conjugacy of words, the construction of centralizers of an element and a finitely generated subgroup.The class of groups considered in the article is a tree product of Artin groups with a tree structure and Artin groups of extra-large type, amalgamated by cyclic subgroups corresponding to the generators of the groups.Artin groups were introduced at the beginning of the last century as a generalization of the well-known braid groups, the class of extra-large Artin groups was isolated in 1983, the class of Artin groups with a woody structure was isolated in 2003. The groups considered in this paper belong to almost large Artin groups and the problems of words and conjugacy of words are algorithmically solvable in them, which follows from the proof of their biautomaticity. Theapproach proposed by the authors in solving the problem of power conjugacy of words is morevisual and simple.

Abstract We present a solution to the conjugacy problem in the group of outer automorphisms of $F_3$ , a free group of rank 3. We distinguish according to several computable invariants, such as irreducibility, subgroups of polynomial growth and subgroups carrying the attracting lamination. We establish, by considerations on train tracks, that the conjugacy problem is decidable for the outer automorphisms of $F_3$ that preserve a given rank 2 free factor. Then we establish, by consideration on mapping tori, that it is decidable for outer automorphisms of $F_3$ whose maximal polynomial growth subgroups are cyclic. This covers all the cases left by the state of the art.

This paper presents a robust lattice-based digital signature scheme that leverages matrix groups to enhance post-quantum security. Built on the hardness of lattice problems such as the Shortest Vector Problem (SVP) and Learning With Errors (LWE), combined with the complexity of the Matrix Group Conjugacy Problem our scheme demonstrates both theoretical and practical security. We rigorously establish the (MGCP), mathematical foundations, analyze the computational complexity, and provide numerical simulations to evaluate performance. This approach contributes a unique blend of lattice and matrix group theory, offering new insights and possibilities in post-quantum cryptography.

We prove that the Brinkmann Problems ( BrP & BrCP ) and the Twisted-Conjugacy Problem ( TCP ) are decidable for any endomorphism of a free-abelian times free (FATF) group [Formula: see text]. Furthermore, we prove the decidability of the two-sided Brinkmann Conjugacy Problem ( 2BrCP ) for monomorphisms of FATF groups (and combine it with TCP ) to derive the decidability of the conjugacy problem for ascending HNN extensions of FATF groups.

On the conjugacy problem for finite groups in the plane Cremona group

In 1971 C. F. Miller associated to every finitely presented group G a free-by-free group [Formula: see text] known as the Miller Machine, whose conjugacy problem is closely related to the conjugacy and word problems of G. We quantify this relationship, and look to fully understand the conjugacy problem of [Formula: see text]; namely, we reduce the conjugacy problem in [Formula: see text] to a strong form of list conjugacy in G, which we term iso-computational list conjugacy. As an application, we show that if G is finite, the conjugacy problem for [Formula: see text] is in [Formula: see text].

We prove that, for any hyperbolic group, the compressed word and the compressed conjugacy problems are solvable in polynomial time. As a consequence, the word problem for the (outer) automorphism group of a hyperbolic group is solvable in polynomial time. We also prove that the compressed simultaneous conjugacy and the compressed centraliser problems are solvable in polynomial time. Finally, we prove that, for any infinite hyperbolic group, the compressed knapsack problem is \mathrm{NP} -complete.

Abstract We prove that centralisers of elements in [finitely generated free]-by-cyclic groups are computable. As a corollary, given two conjugate elements in a [finitely generated free]-by-cyclic group, the set of conjugators can be computed and the conjugacy problem with context-free constraints is decidable. We pose several problems arising naturally from this work.

A method for digital signatures ensures the security of messages sent between individuals. Several algorithms have been developed by focusing on individual challenging issues like conjugacy problem, discrete logarithm problem, and integer factorization problem. Though, it has been noted that these techniques are challenging to calculate in order to get at an appropriate solution. These days, the majority of algorithms are created by combining two challenging tasks. With a single challenging problem, we created an algorithm that deals comparable security. In this work, we provide a novel digital signature scheme that takes advantage of non-commutative rings characteristics. The digital signature is protected by the hardness of the conjugacy problem on non-commutative structures and also it is designed by using differential polynomials. We believe that conjugacy problem is NP-hard. The confirmation theorem was used to show the algorithm's strength. Security analysis was also explained.

We prove that the compressed conjugacy problem in a group that is hyperbolic relative to a collection of free abelian subgroups is solvable in polynomial time.

We establish a connection between the generalized conjugacy problem for a G-by- group, , and two algorithmic problems for G: the generalized Brinkmann’s conjugacy problem, GBrCP(G), and the generalized twisted conjugacy problem, GTCP(G). We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to their cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that GBrCP(G) is decidable (with respect to cosets) when G is a virtually polycyclic group, which implies in particular that the generalized Brinkmann’s equality problem, GBrP(G), is decidable if G is a finitely generated abelian group. Finally, we prove that if G is a finitely generated virtually free group, then the simple versions of Brinkmann’s equality problem and of the twisted conjugacy problem, BrP(G) and TCP(G), are decidable.

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to right-angled Artin groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a right-angled Artin group G, certain Magnus subgroups embed in the quotient G=G/N(r); that if r=sn has root s in G then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. We also give conditions under which the question of which Magnus subgroups of G embed in G reduces to the same question in the minimal parabolic subgroup of G containing r. In many cases this allows us to characterise Magnus subgroups which embed in G, via a condition on r and the commutation graph of G, and to find further examples of quotients G where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord Γ, almost all cyclically reduced elements of the right-angled Artin group G(Γ) satisfy the conditions of the theorem.