For Groups the Property of Having Finite Derivation Type is Equivalent to the Homological Finiteness Condition FP_3

The homological niteness property F P 3 and the combinatorial property of having nite derivation type both are necessary conditions for nitely presented monoids to admit a nite convergent presentation. For monoids in general, the property of having nite derivation type is strictly stronger than the property F P 3. Here we show that for groups these two properties are equivalent. The proof exploits a result of 6], which states that a group G, which is given through a nite presentation hX; Ri, has nite derivation type if and only if the ZG-module of identities among relations that is associated with hX; Ri is nitely generated. We also give a new proof for this result which is much simpler than our original proof.