1999/01/21 by Robert H. Gilman, Gilman, Robert H., Derek F. Holt +3
Mathematics · #20-04 #20F10 #68Q40 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:03D40 #msc:20-04 #msc:20F10 #msc:68Q40 #secondary: 03D40
paper · pdf · doi:10.48550/arxiv.math/9901088
To appear in the International Journal of Algebra and Computation
arxiv created 1999/01/21 · arxiv updated 2009/11/30
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.