Universal groups, binate groups and acyclicity. (English) Zbl 0663.20053

Berrick, A. J.

Group theory, Proc. Conf., Singapore 1987, 253-266 (1989).

[For the entire collection see Zbl 0652.00004.]
The author introduces the notion of a “binate group” \((``binate''=``arranged\) in pairs”). More precisely, the universal binate-filtered group with base H is the tower \(H=H_ 0\leq H_ 1\leq H_ 2\leq..\). where \(H_{i+1}\) is the HNN-extension \((H_ i\times H_ i)*\psi\) resulting from \(\psi\) : [1]\(\times H_ i\to H_ i\times H_ i\) given by \(\psi (l,h)=(h,h)\). It is shown that any binate-filtered group is an infinitely generated, acyclic group.
The author examines the basic properties of binate-ness and uses them to prove acyclicity for a number of groups (e.g. Ph. Hall’s countable, universal locally finite group [J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.023)], groups of diffeomorphisms, mitotic groups, general linear groups of the cone on a ring etc.).

Reviewer: V.P.Snaith

MSC:

20J05	Homological methods in group theory
20E06	Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
55R40	Homology of classifying spaces and characteristic classes in algebraic topology
20F05	Generators, relations, and presentations of groups

Keywords:

universal binate-filtered group; HNN-extension; infinitely generated, acyclic group; universal locally finite group; groups of diffeomorphisms; mitotic groups; general linear groups