FOR GROUPS THE PROPERTY OF HAVING FINITE DERIVATION TYPE IS EQUIVALENT TO THE HOMOLOGICAL FINITENESS CONDITION FP3

The homological finiteness property FP3 and the combinatorial property of having finite derivation type are both necessary conditions for fi nitely presented monoids to admit finite convergent presentations. For monoids in general, the property of having finite derivation type imp lies the property FP3, and there even exist finitely presented monoids that are FP3, but that do not have finite derivation type (Cremanns a nd Otto, 1994). sere, 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 ZG-module of identities among relations that is associated with (X;R) is finitely generated. T his result, which was announced in (Cremanns and Otto, 1994), is prove d in a conceptually simple manner, greatly improving upon the original proof that was only outlined in (Cremanns and Otto, 1994). Then, usin g elementary algebraic arguments we derive our main result without usi ng much of homology theory, thus making the proof easily accessible to computer scientists and mathematicians with some background in algebr a and rewriting theory. (C) 1996 Academic Press Limited