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Generalised Magnus modules over the braid group. (English) Zbl 0859.20030

The present article considers the construction of homology modules over the braid group. A well-known procedure of W. Magnus, restated by J. Birman, is shown to be equivalent to considering the homology \(H_1(F,M)\) of the free group \(F\) with certain coefficient modules \(M\) as braid module. The main result of the work is the construction of a free resolution \(C\) of the integers over an iterated semidirect product of free groups. The complexes \(P\otimes C\) and their homology for certain coefficient modules \(P\) then carry the structure of a braid module. These can be regarded as generalisations of the Burau- and Magnus-modules. The resolution is further used to show that the holonomy representations of the braid group and of the Hecke algebra constructed by topological means by R. J. Lawrence belong to this class of modules.
Reviewer: M.Lüdde (Berlin)

MSC:

20F36 Braid groups; Artin groups
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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References:

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