Automatic Groups and Knuth-Bendix with Infinitely Many Rules

Knuth-Bendix算法与幺半群中的共轭问题

Knuth-Bendix algorithm and the conjugacy problems in monoids

本文针对幺半群共轭问题，扩展Knuth-Bendix经典重写理论，提出纳入循环共轭的重写算法方法。

Knuth-Bendix for groups with infinitely many rules

It is shown how to use a small finite state automaton in two variables in order to carry out the Knuth-Bendix process for rewriting words in a group in shortlex order. The two-variable automaton can be used to store an infinite set of rules and to carry out fast reduction of arbitrary words using this infinite set. We introduce a new operation, which we call welding, which applies to an arbitrary finite state automaton. We show how to improve on the standard subset construction to determinize a non-deterministic automaton under special conditions which hold in our situation.

Infinite presentability of groups and condensation

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor-Bendixson…

Geodesic rewriting systems and pregroups

In this paper we study rewriting systems for groups and monoids, focusing on situations where finite convergent systems may be difficult to find or do not exist. We consider systems which have no length increasing rules and are confluent and then systems in which the length reducing rules lead to geodesics. Combining these properties we arrive at our main object of study which we call geodesically perfect rewriting systems. We show that these are well-behaved and convenient to use, and give several examples of classes of groups for which they can be constructed from natural presentations. We d…

完美核与动力学：从Bass-Serre理论到双曲群

Perfect kernel and dynamics: from Bass-Serre theory to hyperbolic groups

本文通过引入新方法研究可数群子群空间的Cantor-Bendixson分解与完美核，确定了许多新群的完美核和秩，建立了共轭作用的拓扑传递性条件，并由此得到类A中大量新的群例，涵盖直角Artin群、极限群、小消去群、随机群及虚拟紧特殊群等。

Coxeter群中的Bender--Knuth台球

Bender--Knuth Billiards in Coxeter Groups

本文为任意Coxeter群引入不可逆Bender--Knuth切换算子，提出“未来主义”概念并证明多类Coxeter群满足该性质，同时给出了非未来主义群的判定。