Combing nilpotent and polycyclic groups

Robert H. Gilman; Derek F. Holt; Sarah Rees

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Combing nilpotent and polycyclic groups

Robert H. Gilman·Derek F. Holt·Sarah Rees

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Abstract

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 group embeds in a real-time combable group. All the combings constructed in the article are boundedly asynchronous, and those for nilpotent-by-finite groups have polynomially bounded length functions, of degree equal to the nilpotency class, c. This result verifies a polynomial upper bound on the Dehn functions of those groups of degree c+1.

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BibTeX RIS

Robert H. Gilman, Derek F. Holt, Sarah Rees. 1999-01-21. Combing nilpotent and polycyclic groups. https://arxiv.org/abs/math/9901088

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