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An algorithm for computing homology groups. (English) Zbl 0895.20043
In a previous article [in Proc. Canberra Group Theory Conf. 1989, Lect. Notes Math. 1456, 114-141 (1990; Zbl 0732.20032)] the author described a procedure which, for a finite rewriting system of a group \(G\), or even a monoid, yields a free resolution of \(\mathbb{Z}\) by \(\mathbb{Z} G\)-modules. A problem with that and similar constructions is the difficulty of explicitly computing the differential of the resolution. In this paper, the author presents an algorithm for computing the differentials in that resolution.

20J05 Homological methods in group theory
20-04 Software, source code, etc. for problems pertaining to group theory
20F05 Generators, relations, and presentations of groups
20M35 Semigroups in automata theory, linguistics, etc.
68Q42 Grammars and rewriting systems
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[1] Anick, D.J., On the homology of associative algebras, Trans. amer. math. soc., 296, 641-659, (1986) · Zbl 0598.16028
[2] Brown, K.S., Cohomology of groups, (1982), Springer-Verlag New York · Zbl 0367.18012
[3] Brown, K.S., The geometry of rewriting systems: A proof of the anick-groves-squier theorem, (), 137-163 · Zbl 0764.20016
[4] M. Buchheit, 1991, Algorithmen zur Berechnung von Invarianten für konvergente Wortersetzungs-Systeme, Universität Kaiserslautern
[5] Groves, J.R.J., Rewriting systems and homology of groups, Proceedings of the Canberra group theory conference, 1989, Lecture notes in mathematics, 1456, (1990), Springer-Verlag New York/Berlin, p. 114-141 · Zbl 0732.20032
[6] Kobayashi, Y., Complete rewriting systems and homology of monoid algebras, J. pure appl. algebra, 65, 263-275, (1990) · Zbl 0711.20035
[7] Squier, C.C., Word problems and a homological finiteness condition for modules, J. pure appl. algebra, 40, 201-217, (1987) · Zbl 0648.20045
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