Groups with a Complemented Presentation
نویسنده
چکیده
Let G be a group given by a presentation. We study the decomposition of the elements of G as quotients of “positive” elements (the elements of G that can be expressed without using the inverses of the generators) in the special case when the presentation satisfies some syntactical condition. This approach works in particular for Artin’s braid groups, and results in a very simple quadratic algorithm for solving their word problem. AMS Classification: 20M05, 20F36. Artin’s braid group B∞ is the group generated by an infinite sequence σ1, σ2, . . . submitted to the relations {σiσi+1σi = σi+1σiσi+1, σiσj = σjσi for |i− j| ≥ 2. One observes that this presentation has the particular syntactical property that, for any two generators x, y, there exists exactly one relation of the form
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