Abstract
Let G be a group or monoid which is presented by means of a complete rewriting system. Then one can use the resulting normal forms to collapse the classifying space of G down to a quotient complex (typically “small”) of the same homotopy type. If the rewriting system is finite, then the quotient complex has only finitely many cells in each dimension. The proof yields an explicit free resolution of Z over Z G, similar to resolutions obtained by Anick, Groves, and Squier.
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References
D. J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296 (1986), 641–659.
G. M. Bergman, The diamond lemma for ring theory, Advances in Math. 29 (1978), 178–218.
M. G. Brin and C. C. Squier, Groups of piecewise linear homeomorphisms of the real line, Invent. Math. 79 (1985), 485–498.
K. S. Brown, “Cohomology of Groups,” Graduate Texts in Mathematics 87, Springer-Verlag, New York, 1982.
K. S. Brown, Finiteness properties of groups, J. Pure Appl. Algebra 44 (1987), 45–75.
K. S. Brown and R. Geoghegan, An infinite-dimensional torsion-free FP. group, Invent. Math. 77 (1984), 367–381.
H. Cartan and S. Eilenberg, “Homological Algebra,” Princeton University Press, Princeton, 1956.
B. Eckmann, Der Cohomologie-Ring einer beliebigen Gruppe, Comment. Math. Helv. 18 (1946), 232–282.
S. Eilenberg and S. MacLane, Relations between homology and homotopy groups of spaces, Ann. of Math. (2) 46 (1945), 480–509.
Z. Fiedorowicz, Classifying spaces of topological monoids and categories, Amer. J. Math. 106 (1984), 301–350.
J. R. J. Groves, Rewriting systems and resolutions over group rings,in preparation.
J. R. J. Groves and G. C Smith, Rewriting systems and soluble groups,preprint.
D. E. Knuth, “The Art of Computer Programming,” Volume 1, Fundamental algorithms, second edition, Addison-Wesley, Reading, 1973.
P. Le Chenadec, “Canonical Forms in Finitely Presented Algebras,” Research Notes in Theoretical Computer Science, Pitman (London-Boston) and Wiley (New York), 1986.
R. C. Lyndon, Cohomology theory of groups with a single defining relation, Ann. of Math. 52 (1950), 650–665.
D. McDuff, On the classifying spaces of discrete monoids, Topology 18 (1979), 313–320.
D. Quillen, Higher algebraic K-theory. I, in “Algebraic K-theory, I: Higher K-theories,” Proc. Conf., Battelle Memorial Inst., Seattle, 1972, Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147.
G. B. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 105–112.
C. C. Squier, Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49 (1987), 201–217.
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© 1992 Springer-Verlag New York, Inc.
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Brown, K.S. (1992). The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem. In: Baumslag, G., Miller, C.F. (eds) Algorithms and Classification in Combinatorial Group Theory. Mathematical Sciences Research Institute Publications, vol 23. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9730-4_6
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DOI: https://doi.org/10.1007/978-1-4613-9730-4_6
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