For Groups the Property of Having Finite Derivation Type is equivalent to the Homological Finiteness ConditionFP3
✍ Scribed by ROBERT CREMANNS; FRIEDRICH OTTO
- Publisher
- Elsevier Science
- Year
- 1996
- Tongue
- English
- Weight
- 694 KB
- Volume
- 22
- Category
- Article
- ISSN
- 0747-7171
No coin nor oath required. For personal study only.
✦ Synopsis
The homological finiteness property F P 3 and 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 F P 3 , and there even exist finitely presented monoids that are F P 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 ZG-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.