For any group $G$ and integer $k\ge 2$ the Andrews-Curtis transformations act as a permutation group, termed the Andrews-Curtis group $AC_k(G)$, on the subset $N_k(G) \subset G^k$ of all $k$-tuples that generate $G$ as a normal subgroup (provided $N_k(G)$ is non-empty). The famous Andrews-Curtis Conjecture is that if $G$ is free of rank $k$, then $AC_k(G)$ acts transitively on $N_k(G)$. The set $N_k(G)$ may have a rather complex structure, so it is easier to study the full Andrews-Curtis group $FAC(G)$ generated by AC-transformations on a much simpler set $G^k$. Our goal here is to investigate the natural epimorphism $λ\colon FAC_k(G) \to AC_k(G)$. We show that if $G$ is non-elementary torsion-free hyperbolic, then $FAC_k(G)$ acts faithfully on every nontrivial orbit of $G^k$, hence $λ\colon FAC_k(G) \to AC_k(G)$ is an isomorphism.7 pages. In memory of Ben Fine. Published in journal of Groups, Complexity, Cryptology

Abstract The universal pairing for manifolds was defined and shown to lack positivity in dimension 4 in [Freedman, Kitaev, Nayak, Slingerland, Walker, and Wang, J. Geom. Topol. 9 (2005), 2303–2317]. We prove an analogous result for 2‐complexes, and show that the universal pairing does not detect the difference between simple homotopy equivalence and 3‐deformations. The question of whether these two equivalence relations are different for 2‐complexes is the subject of the Andrews–Curtis conjecture. We also discuss the universal pairing for higher dimensional complexes and show that it is not positive.

An exotic presentation of ℤ × ℤ and the Andrews–Curtis conjecture

The first author introduced a notion of equivalence on a family of 3-manifolds with boundary, called (simple) balanced 3-manifolds in an earlier paper and discussed the analogy between the Andrews-Curtis equivalence for group presentations and the aforementioned notion of equivalence. Motivated by the Andrews-Curtis conjecture, we use tools from Heegaard Floer theory to prove that there are simple balanced 3-manifolds which are not in the trivial equivalence class (i.e. the equivalence class of S 2 × [−1, 1]).

New Andrews–Curtis trivializations for Miller–Schupp group presentations

The Andrews-Curtis conjecture, a long-standing problem in algebraic topology, has remained a subject of intense research for decades. In this paper, we introduce a novel reduction method that expands the boundaries of finite topological spaces satisfying the conjecture. Our method builds upon the foundational work in the field and offers a fresh perspective on tackling this challenging problem. Through the application of our innovative reduction technique, we demonstrate the discovery of a substantial number of previously unidentified finite topological spaces that satisfy the Andrews-Curtis conjecture. Additionally, we investigate how our reduction method works together with reduction methods previously introduced by Ximena [7]. This collaborative investigation not only validates the efficacy of our method but also reveals intriguing connections between different reduction techniques, shedding light on the underlying mathematical structures governing the conjecture.

Abstract We develop new computational methods for studying potential counterexamples to the Andrews–Curtis conjecture, in particular, Akbulut–Kurby examples{\operatorname{AK}(n)}. We devise a number of algorithms in an attempt to disprove the most interesting counterexample{\operatorname{AK}(3)}. That includes an efficient implementation of the folding procedure for pseudo-conjugacy graphs, based on the original modification of a classic disjoint-set data structure. To improve metric properties of the search space (the set of balanced presentations of the trivial group), we introduce a new transformation, called an ACM-move, that generalizes the original Andrews–Curtis transformations and discuss details of a practical implementation. To reduce growth of the search space, we introduce a strong equivalence relation on balanced presentations and study the space modulo automorphisms of the underlying free group. We prove that automorphism moves can be applied to Akbulut–Kurby presentations. The improved technique allows us to enumerate balanced presentations AC-equivalent to{\operatorname{AK}(3)}with relations of lengths up to 20 (previous record was 17).

The Andrews–Curtis conjecture claims that every normally generating [Formula: see text]-tuple of a free group [Formula: see text] of rank [Formula: see text] can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing [Formula: see text] by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews–Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag–Solitar groups do not satisfy the generalized Andrews–Curtis conjecture in the sense of Burns and Macedońska.

Abstract The Andrews–Curtis conjecture asserts that, for a free group Fn of rank n and a free basis (x 1,...,xn ), any normally generating tuple (y 1,...,yn ) is Andrews–Curtis equivalent to (x 1,...,xn ). This equivalence corresponds to the actions of Aut Fn and of F n n on normally generating n-tuples. The equivalence corresponding to the action of Aut Fn on generating n-tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated groups has its own importance to analyse potential counter-examples to the original conjecture. We study the Andrews–Curtis and Nielsen equivalence in the class of finitely generated groups for which every maximal subgroup is normal, including nilpotent groups and Grigorchuk groups.

This article details a series of computational group theory experiments involving a search for irreducible cyclic presentations of the trivial group. The list of such presentations obtained provides test cases for the Andrews–Curtis conjecture.

Abstract.An epimorphism ϕ: G → H of groups, where G has rank n, is called coessential if every (ordered) generating n-tuple of H can be lifted along ϕ to a generating n-tuple for G. We discuss this property in the context of the category of groups, and establish a criterion for such a group G to have the property that its abelianization epimorphism G → G/[G, G], where [G,G] is the commutator subgroup, is coessential. We give an example of a family of 2-generator groups whose abelianization epimorphism is not coessential. This family also provides counterexamples to the generalized Andrews–Curtis conjecture.

We show that a compact n n -polyhedron PL embeds in a product of n n trees if and only if it collapses onto an ( n − 1 ) (n-1) -polyhedron. If the n n -polyhedron is contractible and n ≠ 3 n\ne 3 (or n = 3 n=3 and the Andrews–Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding. In contrast, there exists a 2 2 -dimensional compact absolute retract X X such that X × I k X\times I^k does not embed in any product of 2 + k 2+k dendrites for each k k .

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterize the standard contact structure on the 5-sphere in terms of such fillings. This characterization is discussed in the context of the Andrews–Curtis conjecture concerning presentations of the trivial group. We further illustrate the use of such diagrams by a covering theorem for simply connected spin 5-manifolds and a new existence proof for contact structures on simply connected 5-manifolds.

Attempts have been made to eliminate some potential counterexamples to the Andrews–Curtis conjecture using the combinatorial optimization methods of blind-search and the genetic algorithms meta-heuristic. Breadth-first search with secondary storage is currently the most successful method, which raises questions regarding the inferior performance of heuristic search. In order to understand the underlying reasons we obtain fitness landscape metrics for a number of balanced presentations and draw conclusions regarding the likely effectiveness of other meta-heuristics.

For an integer n at least two and a positive integer m , let C( n,m ) denote the group of Andrews–Curtis transformations of rank ( n,m ) and let F denote the free group of rank n + m . A subgroup AC( n,m ) of Aut(F) is defined, and an anti-isomorphism AC( n,m ) to C( n,m ) is described. We solve the generalized word problem for AC( n,m ) in Aut(F) and discuss an associated reformulation of the Andrews–Curtis conjecture.

In 1979, Rourke proposed to extend the set of cyclically reduced defining words of a group presentation P \mathcal P by using operations of cyclic permutation, inversion and taking double products. He proved that iterations of these operations yield all cyclically reduced words of the normal closure of defining words of P \mathcal P if the group, defined by the presentation P \mathcal P , is trivial. We generalize this result by proving it for every group presentation P \mathcal P with an obvious exception. We also introduce a new, “cyclic", version of the Andrews–Curtis conjecture and show that the original Andrews–Curtis conjecture with stabilizations is equivalent to its cyclic version.

The paper discusses the Andrews–Curtis graph Δk(G,N) of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one of them can be obtained from another by certain elementary transformations. This object appears naturally in the theory of black box finite groups and in the Andrews–Curtis conjecture in algebraic topology [3].We suggest an approach to the Andrews–Curtis conjecture based on the study of Andrews–Curtis graphs of finite groups, discuss properties of Andrews–Curtis graphs of some classes of finite groups and results of computer experiments with generation of random elements of finite groups by random walks on their Andrews–Curtis graphs.

We inititate the systematic study of Artin Presentations, (discovered in 1975 by González-Acuña), which characterize the fundamental groups of closed, orientable 3-manifolds, and form a discrete equivalent of the theory of open book decompositions with planar pages of such manifolds. We list and prove the basic properties, state some fundamental problems and describe some of the advantages of the theory: e.g., an Artin Presentation of π1 (M3) does not just determine the closed, orientable 3-manifold M3, but also a canonical, smooth simply-connected cobordism of it, allowing us to tap into 4-dimensional gauge theory (and 3 + 1 TQFT's) in a more direct, purely discrete, functorial manner than others. Thus, in section 4, instead of using PDE's, we show how a canonical action of the commutator subgroup [Pn, Pn] of the pure braid group Pn can be used to study the smooth structures on a closed, smooth-connected 4-manifold with b2 = n, in a systematic way. However, the main purpose of this first paper is to Artin Presentations to set up simple criteria, testable with, say, MAGMA on the computer (where then no knowledge of topology is required) for finding explicit counter-examples to the so-called Weak Poincaré Conjecture: "Every homotopy 3-sphere bounds a smooth, compact, contractible 4-manifold," as well as: "Every irreducible Z-homology 3-sphere Σ, with π1 (Σ) = I (120) is homeomorphic to Σ (2, 3, 5)" and other conjectures implied by Thurston's Geometrization Conjecture. One first philosophical goal is to convince the reader that the truth of these conjectures is at least as unlikely as that of the Andrews-Curtis Conjecture and that ultimately, Artin Presentation Theory is a non-trivial intersection of string/M theory number theory.

The Andrews–Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with an arbitrary conjugation of a relator. It is believed that the Andrews–Curtis conjecture is false; however, not so many possible counterexamples are known. It is not a trivial matter to verify whether the conjecture holds for a given balanced presentation or not. The purpose of this paper is to describe some nondeterministic methods, called Genetic Algorithms, designed to test the validity of the Andrews–Curtis conjecture. Using such algorithm we have been able to prove that all known (to us) balanced presentations of the trivial group where the total length of the relators is at most 12 satisfy the conjecture. In particular, the Andrews–Curtis conjecture holds for the presentation [Formula: see text] which was one of the well known potential counterexamples.

Grigorchuk and Kurchanov's conjecture states that, up to natural equivalence, there is precisely one splitting epimorphism F2n→Fn×Fn from a free group F2n onto Fn×Fn. The [Formula: see text]-conjecture, while formally similar to the the Poincaré conjecture, implies the well-known Andrews–Curtis conjecture about the balanced presentations of the trivial group. In this paper we undertake the task of fully analyzing the [Formula: see text]-conjecture modulo the second commutator subgroup F2n″. We derive some consequences for the general [Formula: see text]-conjecture.