For Groups the Property of Having Finite Derivation Type is equivalent to the Homological Finiteness Condition FP3

Abstract

The homological finiteness property FP 3and the combinatorial property of having finite derivation type are both necessary conditions for finitely presented monoids to admit finite convergent presentations. For monoids in general, the property of having finite derivation type implies the property FP 3, and there even exist finitely presented monoids that are FP 3, but that do not have finite derivation type (Cremanns and Otto, 1994). Here, contrasting this result, we show that for groups these two properties are equivalent. The proof is based on the result that a group G, which is given through a finite presentation 〈 X; R〉 has finite derivation type if and only if the Z G-module of identities among relations that is associated with 〈 X; R〉 is finitely generated. This result, which was announced in (Cremanns and Otto, 1994), is proved in a conceptually simple manner, greatly improving upon the original proof that was only outlined in (Cremanns and Otto, 1994). Then, using elementary algebraic arguments we derive our main result without using much of homology theory, thus making the proof easily accessible to computer scientists and mathematicians with some background in algebra and rewriting theory.