J Austral Math Soc Ser A 52 pp205--218, 1992.

The (Co)Homology of Groups Given by Presentations in which each Defining Relator Involves at most Two Types of Generators

Stephen J. Pride

Abstract

Our set-up will consist of the following: (i) a graph with vertex set V and edge set E; (ii) for each vertex v Î V a non-trivial group G_v given by a presentation áx_v ; r_vñ; (iii) for each edge e = { u, v} Î E a group G_e given by a presentation áx_u, x_v ; r_eñ where r_e consists of the elements of r_u Èr_v together with some further words on x_u Èx_v. We let G = áx ; rñ where x = È_{v Î V}x_v, r = È_{e Î E}e_e. Our aim is to try to describe the structure of G in terms of the groups G_v (v Î V), G_e (e Î E). Under suitable conditions the natural homomorphisms G_v ® G (v Î V), G_e \rightarrrow G (e Î E) are injective; and there is a short exact sequence

0 ® Å v Î V  (ZG ÄGv IGv)|Star(v)|-1 ® Å e Î E  (ZG ÄGv IGe) ® IG ® 0

(where, for any group H, IH is the augmentation ideal). Some (co)homological consequences of these results are derived.

1991 AMS Subject Classification: 20F05, 20J05

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Authors

Stephen J. Pride

University Gardens, University of Glasgow, Glasgow G12 9QW, Scotland UK.