The Andrews-Curtis conjecture states that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of the elementary Nielsen transformations and conjugations. In this paper we describe all balanced presentations of the trivial group on two generators and with the total length of relators <= 12. We show that all these presentations satisfy the Andrews-Curtis conjecture.
The paper discusses the Andrews-Curtis graph 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 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. 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 fini…
We relate the Andrews-Curtis conjecture to the triviality problem for balanced presentations of groups using algorithms from 3-manifold topology. Implementing this algorithm could lead to counterexamples to the Andrews-Curtis conjecture.
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 non-deterministic methods, called Genetic Algorit…
The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ 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…
The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent importance for computational group theory. It also resolves a question asked in [5] and shows that a computation in finite groups cannot lead to a counterexample to the classical conjecture, as suggested in [5].
It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its "cyclic" version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis made in 1966. We also consider a more restrictive "cancellative" version of the cyclic Andrews--Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews--Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the …
The Andrews-Curtis conjecture asserts that, for a free group $F_n$ of rank $n$ and a free basis $(x_1,...,x_n)$, any normally generating tuple $(y_1,...,y_n)$ is Andrews-Curtis equivalent to $(x_1,...,x_n)$. This equivalence corresponds to the actions of $\operatorname{Aut}F_n$ and of $F_n$ on normally generating $n$-tuples. The equivalence corresponding to the action of $\operatorname{Aut}F_n$ on generating $n$-tuples is called Nielsen equivalence. The conjecture for arbitrary finitely generated group has its own importance to analyse potential counter-examples to the original conjecture. We …
Motivated by problems in topology, we explore the complexity of balanced group presentations. We obtain large lower bounds on the complexity of Andrews-Curtis trivialisations, beginning in rank 4. Our results are based on a new understanding of how Dehn functions of groups behave under certain kinds of push-outs. We consider groups $S$ with presentations of deficiency 1 satisfying certain technical conditions and construct balanced group presentations $¶_w$ indexed by words $w$ in the generators of $S$. If $w=1$ in $S$ then $¶_w$ is Andrews-Curtis trivialisable and the number of Andrews-Curtis…
We develop new computational methods for studying potential counterexamples to the Andrews-Curtis conjecture, in particular, Akbulut-Kurby examples AK(n). We devise a number of algorithms in an attempt to disprove the most interesting counterexample AK(3). To improve metric properties of the search space (which is a set of balanced presentations of the trivial group) we introduce a new transformation (called an ACM-move here) 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…
The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establis…
We pose a representation-theoretic question motivated by an attempt to resolve the Andrews-Curtis conjecture. Roughly, is there a triangular Hopf algebra with a collection of self-dual irreducible representations $V_i$ so that the product of any two decomposes as a sum of copies of the $V_i$, and $\sum (\rank V_i)^2=0$? This data can be used to construct a `topological quantum field theory' on 2-complexes which stands a good chance of detecting counterexamples to the conjecture.
We prove that the presentations $\langle x,y | [x,y],1 \rangle$ and $\langle x,y | [x,[x,y^{-1}]]^2y[y^{-1},x]y^{-1},[x,[[y^{-1},x],x]] \rangle$ are not $Q^*$-equivalent even though their standard complexes have the same simple homotopy type.
We classify all non-collapsing Curtis-Tits and Phan amalgams with $3$-spherical diagram over all fields. In particular, we show that amalgams with spherical diagram are unique, a result required by the classification of finite simple groups. We give a simple condition on the amalgam which is necessary and sufficient for it to arise from a group of Kac-Moody type. This also yields a definition of a large class of groups of Kac-Moody type in terms of a finite presentation.
Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. Th…
Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by …
For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can …
What makes math problems hard for reinforcement learning: a case study
Western University（西安大略大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）；Institute of Mathematics, University of Warsaw（华沙大学数学研究所）；Polygon Zero；Richard N. Merkin Center for Pure and Applied Mathematics, California Institute of Technology（加州理工学院理查德·N·默kin纯粹与应用数学中心）
