A language theoretic analysis of combings

Hairdressing in groups: a survey of combings and formal languages

A group is combable if it can be represented by a language of words satisfying a fellow traveller property; an automatic group has a synchronous combing which is a regular language. This article surveys results for combable groups, in particular in the case where the combing is a formal language.

Combing nilpotent and polycyclic groups

A combing is a set of normal forms for a finitely generated group. This article investigates the language-theoretic and geometric properties of combings for nilpotent and polycyclic groups. It is shown that a finitely generated class 2 nilpotent group with cyclic commutator subgroup is real-time combable, as are also all 2 or 3-generated class 2 nilpotent groups, and groups in certain families of nilpotent groups, e.g. the finitely generated Heisenberg groups, groups of unipotent matrices over the integers and the free class 2 nilpotent groups. Further it is shown that any polycyclic-by-finite…

Multiplication tables and word-hyperbolicity in free products of semigroups, monoids, and groups

This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…

Formal Languages and Infinite Groups

This article is an introduction to formal languages from the point of view of combinatorial group theory. Group theoretic applications are included and language classes are defined algebraically.

从语言学角度研究自由群中子集的密度

A language-theoretic approach to study the density of subsets in free groups

本文通过语言相对密度研究自由群中子集的密度，证明了语言密度的性质，并用语言学方法证明了非交换自由群中元素的自同构轨道自然密度为零，同时探讨了有限生成子群的正密度特性。