Complete positive group presentations
By Patrick Dehornoy. 2001
61 citations recorded by the source corpus.
Abstract A combinatorial property of positive group presentations, called completeness, is introduced, with an effective criterion for recognizing complete presentations, and an iterative method for completing an incomplete presentation. We show how to directly read several properties of the associated monoid and group from a complete presentation: cancellativity or existence of common multiples in the case of the monoid, or isoperimetric inequality in the case of the group. In particular, we obtain a new criterion for recognizing that a monoid embeds in a group of fractions. Typical presentations eligible for the current approach are the standard presentations of the Artin–Tits groups and the Heisenberg group.
Subfields: Computational Theory and Mathematics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/2187ecc52555ee101553a3922a6bbd766ef3bc7e
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In Computational Theory and Mathematics
In Geometry and Topology
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- Eric Jespers
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- Ferran Cedó
- Tatiana Gateva-Ivanova
Citations in the corpus
- Monoids and Groups of I-Type (2005, methodology): “Because of the results proved in [ 4 ], this yields another proof of the fact that S is cancellative and has a torsion-free group G of fractions, but also implies some other properties of G, for example, that G has a solvable word problem.”
- The group of parenthesized braids (2004, unknown): “In order to study the group B̃•, we resort to general algebraic tools developed in [14, 16] and connected with Garside’s seminal work [22].”
- Garside Groups and Yang–Baxter Equation (2009, unknown): “They also re-mark that the defining presentation of a monoid of I-type satisfies the right cube condition, as defined by Dehornoy in [7, Prop.4.4].”
- Partial transformation groupoids attached to graphs and semigroups (2016, background): “Let P = 〈S,R〉 be a monoid given by a positive r-complete presentation (S,R) in the sense of [7].”