By Michael D. Atkinson

ISBN-10: 0120662701

ISBN-13: 9780120662708

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B. Renz, Geometric invariants and HNN-extensions, in: Group Theory, Proceedings of Singapore conference (1987), de Gruyter, Berlin, 1989, pp. 465-484. B. Renz, Geometrische Invarianten and Endlichkeitseigenschaften von Gruppen, Dissertation, Frankfurt, 1988. J: C. Sikorav, Homologie de Novikov associee a une classe de cohomologie rdelle de degre un, preprint, 1989. R. Strebel, Finitely presented soluble groups, in: Group Theory - Essays for Philip Hall (K. W. Gruenberg, J. E. ), Academic Press, 1984, pp.

But I feel it is worthwhile to give an explicit algebraic translation of his proof, so that such readers do not have to take his main theorem on trust. We begin with a chain complex of free modules over a ring A. We require not only that each P is a free module but also that a specific basis is chosen for each Pn; the elements of this basis are called n-cells. A collapsing scheme on P is defined to consist of the following: (1) a division of the cells into three pairwise disjoint classes, which we refer to as the essential, redundant, and collapsible cells, with all 0-cells being essential and all 1-cells being either essential or redundant; (2) a function, called weight, from the set of all redundant cells into N; (3) a bijection, for each n, between the set of redundant n-cells and the set of collapsible (n + 1)-cells, such that, if the collapsible cell c corresponds to the redundant cell r, then there is a unit u of A for which all redundant cells in the chain r - uOc have weight less than the weight of r (in particular, if r has weight 0 then r - uac contains no redundant cells).

But it is most instructive to translate, at least intuitively, the concepts of essential connectivity and horo-connectivity. Clearly Xh should be called essentially k-acyclic if there is a real number d > 0 such that inclusion induces the zero map d Hi(Xh) - Hi(Xh -))' for all i < k (reduced homology). To translate horo-connectedness into horoacyclicity is more interesting: Corresponding to continuous maps f : R' -> X we consider infinite locally finite i-chains of X (ordinary i-chains would correspond to continuous maps Si - X).

### Computational Group Theory: Proceedings of the London Mathematical Society Symposium on Computational Group Theory by Michael D. Atkinson

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