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The (CO)homology of groups given by presentations in which each defining relator involves at most two types of generators

Part of: Connections with homological algebra and category theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Stephen J. Pride
Stephen J. Pride
Affiliation:
University GardensUniversity of GlasgowGlasgow G12 8QW Scotland, UK
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Abstract

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Our set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex ∈ V a non-trivial group Gv given by a presentation (xν; rν); (iii) for each edge e = {u, ν} ∈ E a group Ge given by a presentation (xu, xv; re) where re consists of the elements of ru ∪ rv, together with some further words on xu ∪ xv. We let G = (x; r) where x = ∪v∈v xv, r = ∪e∈E re. Ouraim is to try to describe the structure of G in terms of the groups Gv, (v ∈ V), Ge (e ∈ E). Under suitable conditions the natural homomorphisms Gv, → G (ν ∈ V), Ge → Ge (e ε E) are injective; and there is a short exact sequence (where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these resultsare derived.

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20J05: Homological methods in group theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 2 , April 1992 , pp. 205 - 218
Copyright
Copyright © Australian Mathematical Society 1992

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