On the Andrews-Curtis conjecture and algorithms from topology

On Andrews-Curtis conjectures for soluble groups

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 Finitary Andrews-Curtis Conjecture

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].

Andrews-Curtis and Nielsen equivalence relations on some infinite groups

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 …

A counterexample to a strong version of the Andrews-Curtis 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.

Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture

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.

通过4-流形三分解实现的一族Andrews-Curtis平凡化

A family of Andrews-Curtis trivializations via 4-manifold trisections

本文利用4-流形三分解技术，证明了由R-链环族 $L(3,2;4/d)$ 对应的所有群展示对所有 $d$ 均是Andrews-Curtis平凡的。

Conjugacy search problem and the Andrews-Curtis conjecture

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…

Genetic algorithms and 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…

Representation theory, topological field theory, and the Andrews-Curtis conjecture

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.

关于 Andrews-Curtis 群：非有限展示性

On the Andrews-Curtis groups: non-finite presentability

本文定义了一类广义 Andrews-Curtis 群 $\{A_{r,s}\}$，并证明 $A_{2,s}$（$s \geq 1$）是非有限展示的，从而得出 AC($F_2$) 是非有限展示的，部分回答了关于其有限展示性的著名问题。

Andrews-Curtis 群

Andrews-Curtis groups

本文研究 Andrews-Curtis 群，证明对非初等无挠双曲群，完全 Andrews-Curtis 群到 Andrews-Curtis 群的自然满态射是同构，从而简化了相关猜想的研究。

重新审视AK(3)的稳定Andrews-Curtis平凡化：基于自动演绎的案例研究

Stable Andrews-Curtis trivialization of AK(3) revisited. A case study using automated deduction

本文使用自动定理证明器Prover9为平凡群的AK(3)平衡表示与平凡表示的稳定AC等价性提供了替代证明，排除了其作为稳定Andrews-Curtis猜想反例的可能，并提出开发相关计算搜索方法的挑战。

在长度为14的安德鲁斯 - 柯蒂斯边界处的机器可验证等价证书

Machine-checkable equivalence certificates at the length-14 Andrews-Curtis frontier

研究安德鲁斯 - 柯蒂斯猜想在长度为14时的等价性，通过证明六个困难表示中的四个AC等价性得到机器可验证证书，还进行相关分析和程序推导，实现了部分等价性并补充了相关分析，使MS(3) 分支无条件坍缩。

什么让数学问题对强化学习来说如此困难：一项案例研究

What makes math problems hard for reinforcement learning: a case study

该研究以组合群论中的Andrews-Curtis猜想为案例，探索强化学习寻找高奖励罕见实例的挑战，提出算法增强与拓扑硬度度量，还解决了Akbulut-Kirby和Miller-Schupp系列中的数学问题。