Groupoids and the algebra of rewriting in group presentations
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Groupoids and the algebra of rewriting in group presentations N. D. Gilbert1
· E. A. McDougall1
Received: 21 October 2019 / Accepted: 7 September 2020 © The Author(s) 2020
Abstract Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form. Keywords Presentation · Groupoid · Crossed module Mathematics Subject Classification 20F05 · 20J05 · 20L05
Introduction The study of the relationships between presentations of semigroups, monoids, and groups, and systems of rewriting rules has drawn together concepts from group and semigroup theory, low-dimensional topology, and theoretical computer science. Squier (1987) addressed the question of whether a finitely presented monoid with solvable word problem is necessarily presented by a finite, complete, string rewriting system. He proved that a monoid presented by a finite, complete, string rewriting system must satisfy the homological finiteness condition F P3 : indeed, an earlier result of Anick (1986) implies that such a monoid satisfies the stronger condition F P∞ . These ideas are concisely surveyed in Cohen (1993), and more extensively in Otto and Kobayashi (1997). Since examples are known of finitely presented monoids with solvable word problem that do not satisfy F P3 , Squier’s work shows that such monoids need not be presented by finite, complete, string rewriting systems.
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N. D. Gilbert [email protected] Department of Mathematics and The Maxwell Institute for the Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
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Beitr Algebra Geom
Squier et al. (1994) studied finite, complete, string rewriting systems for monoids and proved that the existence of such a system presenting a monoid M implies a homotopical property—finite derivation type—defined for a graph that encodes the rewriting system. Moreover, they show that having finite derivation type does not depend on the particular rewriting system used to present M, and so is a property of M itself and a necessary condition that M should be presented by a finite, complete string rewriting system. Finite derivation type is naturally thought of as a property of a 2-complex, the Squier complex associated to a monoid presentation P, and obtained by adjoining certain 2-cells to the graph of Squier et al. (1994). This point of view was introduced independently by Pride (1995) and Kilibarda (1997), and then extensively developed in Guba and Sapir (1997, 2006) in terms of both string-rewriting systems, and more geometrically, in terms of directed 2-complexes. The theory developed by Kilibarda and then by Guba and Sapir focusses on the properties of diagram groups, which are fundamental groups
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