For groups the property of having finite derivation type is equivalent to the homological finiteness condition \(FP_ 3\)
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.
combinatorial finiteness properties of finitely presented monoids
finite convergent presentations of groups
scientific article; zbMATH DE number 987594