# zbMATH — the first resource for mathematics

Homology of Gaussian groups. (English) Zbl 1100.20036
Summary: We describe new combinatorial methods for constructing explicit free resolutions of $$\mathbb{Z}$$ by $$\mathbb{Z} G$$-modules when $$G$$ is a group of fractions of a monoid where enough least common multiples exist (“locally Gaussian monoid”), and therefore, for computing the homology of $$G$$. Our constructions apply in particular to all Artin-Tits groups of finite Coxeter type. Technically, the proofs rely on the properties of least common multiples in a monoid.

##### MSC:
 20J05 Homological methods in group theory 20M05 Free semigroups, generators and relations, word problems 20F36 Braid groups; Artin groups 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20M50 Connections of semigroups with homological algebra and category theory
Full Text:
##### References:
 [1] Fragments of the word delta in a braid group, Mat. Zam. Acad. Sci. SSSR ; transl. Math. Notes Acad. Sci. USSR, 36 ; 36, 1-1, 25-34 ; 505-510, (19841984) · Zbl 0599.20044 [2] A geometric rational form for Artin groups of FC type, Geometriae Dedicata, 79, 277-289, (2000) · Zbl 1048.20020 [3] The cohomology ring of the colored braid group, Mat. Zametki, 5, 227-231, (1969) · Zbl 0277.55002 [4] Toplogical invariants of algebraic functions II, Funkt. Anal. Appl., 4, 91-98, (1970) · Zbl 0239.14012 [5] The dual braid monoid · Zbl 1064.20039 [6] Non-positively curved aspects of Artin groups of finite type, Geometry & Topology, 3, 269-302, (1999) · Zbl 0998.20034 [7] A new approach to the word problem in the braid groups, Advances in Math., 139, 2, 322-353, (1998) · Zbl 0937.20016 [8] Sur LES groupes de tresses (d’après V.I. Arnold), Sém. Bourbaki, exp. no 401 (1971), 317, 21-44, (1973) · Zbl 0277.55003 [9] Artin-gruppen und Coxeter-gruppen, Invent. Math., 17, 245-271, (1972) · Zbl 0243.20037 [10] Cohomology of groups, (1982), Springer · Zbl 0584.20036 [11] Homological Algebra, (1956), Princeton University Press, Princeton · Zbl 0075.24305 [12] Artin groups of finite type are biautomatic, Math. Ann., 292, 4, 671-683, (1992) · Zbl 0736.57001 [13] Geodesic automation and growth functions for Artin groups of finite type, Math. Ann., 301, 2, 307-324, (1995) · Zbl 0813.20042 [14] Bestvina’s normal form complex and the homology of Garside groups · Zbl 1064.20044 [15] The algebraic theory of semigroups, vol. 1, AMS Surveys, 7, (1961) · Zbl 0111.03403 [16] Cohomology of braid spaces, Bull. Amer. Math. Soc., 79, 763-766, (1973) · Zbl 0272.55012 [17] Artin’s braid groups, classical homotopy theory, and sundry other curiosities, Contemp. Math., 78, 167-206, (1988) · Zbl 0682.55011 [18] Cohomology of Artin groups, Math. Research Letters, 3, 296-297, (1996) · Zbl 0870.57004 [19] The top-cohomology of Artin groups with coefficients in rank 1 local systems over Z, Topology Appl., 78, 1, 5-20, (1997) · Zbl 0878.55003 [20] Deux propriétés des groupes de tresses, C. R. Acad. Sci. Paris, 315, 633-638, (1992) · Zbl 0790.20056 [21] Gaussian groups are torsion free, J. of Algebra, 210, 291-297, (1998) · Zbl 0959.20035 [22] Braids and self-distributivity, vol. 192, (2000), Birkhäuser · Zbl 0958.20033 [23] Groupes de garside, Ann. Sci. École Norm. Sup., 35, 267-306, (2002) · Zbl 1017.20031 [24] Complete group presentations · Zbl 0862.20025 [25] Gaussian groups and garside groups, two generalizations of Artin groups, Proc. London Math. Soc., 79, 3, 569-604, (1999) · Zbl 1030.20021 [26] LES immeubles des groupes de tresses généralisés, Invent. Math., 17, 273-302, (1972) · Zbl 0238.20034 [27] Algorithms for positive braids, Quart. J. Math. Oxford, 45, 2, 479-497, (1994) · Zbl 0839.20051 [28] Word Processing in Groups, (1992), Jones & Bartlett Publ. · Zbl 0764.20017 [29] Cohomology of the braid group mod. 2, Funct. Anal. Appl., 4, 143-151, (1970) · Zbl 0222.57031 [30] The braid group and other groups, Quart. J. Math. Oxford, 20, 78, 235-254, (1969) · Zbl 0194.03303 [31] The cohomology of braid groups of series C and D and certain stratifications, Funkt. Anal. i Prilozhen., 12, 2, 76-77, (1978) · Zbl 0409.20032 [32] Complete rewriting systems and homology of monoid algebras, J. Pure Appl. Algebra, 65, 263-275, (1990) · Zbl 0711.20035 [33] A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. squier), J. Pure Appl. Algebra, 98, 229-244, (1995) · Zbl 0832.20080 [34] Church-rosser property and homology of monoids, Math. Struct. Comput. Sci, 1, 297-326, (1991) · Zbl 0748.68035 [35] Higher syzygies, in “une dégustation topologique: homotopy theory in the swiss alps”, Contemp. Math., 265, 99-127, (2000) · Zbl 0978.20022 [36] Petits groupes gaussiens, (2000) [37] The center of thin Gaussian groups, J. Algebra, 245, 1, 92-122, (2001) · Zbl 1002.20022 [38] Topology of the complement of real hyperplanes in $$\mathbf C^N,$$ Invent. Math., 88, 3, 603-618, (1987) · Zbl 0594.57009 [39] The homotopy type of Artin groups, Math. Res. Letters, 1, 565-577, (1994) · Zbl 0847.55011 [40] Extraction of roots in garside groups, Comm. in Algebra, 30, 6, 2915-2927, (2002) · Zbl 1007.20036 [41] Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra, 49, 201-217, (1987) · Zbl 0648.20045 [42] The homological algebra of Artin groups, Math. Scand., 75, 5-43, (1995) · Zbl 0839.20065 [43] A finiteness condition for rewriting systems, revision by F. otto and Y. Kobayashi, Theoret. Compt. Sci., 131, 271-294, (1994) · Zbl 0863.68082 [44] The cohomology of pregroups, conference on group theory, Lecture Notes in Math., 319, 169-182, (1973) · Zbl 0263.18016 [45] Finite state algorithms for the braid group, Circulated notes, (1988) [46] Cohomologies of braid groups, Functional Anal. Appl., 12, 135-137, (1978) · Zbl 0424.55015 [47] Braid groups and loop spaces, Uspekhi Mat. Nauk, 54, 2, 3-84, (1999) · Zbl 1124.20304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.