For groups the property of having finite derivation type is equivalent to the homological finiteness condition \(FP_ 3\)
type: Item
instance of: scholarly article
title: For groups the property of having finite derivation type is equivalent to the homological finiteness condition \(FP_ 3\)
zbMATH Open document ID: 0874.20019
author: Robert Cremanns
author: Friedrich Otto
published in: Journal of Symbolic Computation
publication date: 1997-09-25T00:00:00Z
review text: The search for syntactically simple algorithms for solving standard combinatorial problems evolving from a presentation of a monoid such as the word problem, the problem of deciding commutativity and the conjugacy problem led to the discovery of various properties of rewriting systems. In particular, a (string) rewriting system is called convergent if there is no infinite sequence of reductions, and the word problem for a finite convergent rewriting system can be solved by rewriting. This leads to the concept of a finite convergent presentation, and the homological finiteness property \(FP_3\) and the combinatorial property of having what is called finite derivation type (not reproduced here) are both necessary conditions for a finitely presented monoid to admit such a finite convergent presentation. The property of having finite derivation type actually implies the property \(FP_3\) but an earlier result of the authors says that the converse is not true even for finitely presented monoids. It is shown in the paper that for groups the two notions are equivalent. The proof relies on the fact that a finitely presented group has finite derivation type if and only if its module of identities among relations in the sense of Reidemeister and Peiffer [\textit{K. Reidemeister}, Abh. Math. Semin. Univ. Hamb. 16, No. 3/4, 114-118 (1949; Zbl 0040.15001); \textit{R. Peiffer}, Math. Ann. 121, 67-99 (1949; Zbl 0040.15002)] is finitely generated as a module over the group ring.
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 68Q42
Mathematics Subject Classification ID: 20M05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 987594
zbMATH Keywords: combinatorial finiteness properties of finitely presented monoids
zbMATH Keywords: finite derivation type
zbMATH Keywords: finite convergent presentations of groups
zbMATH Keywords: word problem
zbMATH Keywords: conjugacy problem
zbMATH Keywords: convergent string rewriting systems
zbMATH Keywords: identities among relations
zbMATH Keywords: finitely presented groups
reviewed by: J. Huebschmann
MaRDI profile type: Publication
full work available at URL: jsco.1996.0046
OpenAlex ID: W2035318451
DOI: 10.1006/JSCO.1996.0046
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description: scientific article; zbMATH DE number 987594