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Finiteness properties of groups. (English) Zbl 0613.20033
This paper gives some new necessary and sufficient conditions for a group to be of type $$FP_ n$$ (and related conditions for finite presentation). Let X denote a CW-complex and G a group acting on X by homeomorphisms which permute the cells. We say that X is n-good for G if X is acyclic in dimensions less than n and if the stabilizer of any p-cell (p$$\leq n)$$ is of type $$FP_{n-p}$$. (For example, the universal cover of a K(G,1)- complex is good for G.) The author begins by observing the following (well-known) fact.
If G admits an n-good complex X such that X has a finite n-skeleton modulo G, then G is of type $$FP_ n.$$
The conditions the author gives are related to this. If D is a directed set, then a filtration of X is a family of G-invariant subcomplexes of X, indexed by D, such that the union of the subcomplexes is X and $$X_ a\supseteq X_ b$$ if $$a\geq b$$ for a and b in D. The filtration is said to be of finite n-type if each subcomplex $$X_ a$$ has a finite n- skeleton modulo G. Then the following is proved.
Suppose X is n-good with a filtration $$\{X_ a\}$$ of finite n-type. Then G is of type $$FP_ n$$ if and only if, for each $$i<n$$, and for each a in D, there is b in D with $$b\geq a$$ such that the induced map $$H_ j(X_ a)\to H_ j(X_ b)$$ is trivial. Similar results are proved for finite presentation.
The majority of the paper, however, is taken up with applications of these criteria. In particular he considers, groups of the type first studied by R. J. Thompson in 1965, and later generalized by G. Higman [”Finitely presented infinite simple groups”, Notes Pure Math. 8 (1974), Aust. Natl. Univ., Canberra], which yield examples of finitely presented infinite simple groups. He defines several infinite families of such groups, using his criterion to show that all the groups in these families are finitely presented and of type $$FP_{\infty}$$. He also shows that all the groups G in question satisfy $$H^*(G,{\mathbb{Z}}G)=0$$. Some of these facts are in an earlier paper of the author and R. Geoghegan [Invent. Math. 77, 367-381 (1984; Zbl 0557.55009)]. The treatment of these groups is self contained and includes some discussion of their interpretation as groups of homeomorphisms and of their simplicity (or, at least, the simplicity of large subgroups).
The next application is to groups considered by C. H. Houghton [Arch. Math. 31, 254-258 (1978; Zbl 0377.20044)]. These form an infinite family $$\{H_ n\}$$ of infinite permutation groups. The earlier criterion is used to show that the group $$H_ n$$ is of type $$FP_{n-1}$$ but not of type $$FP_ n$$. Finally the family of matrix groups studied by the author and H. Abels [”Finiteness properties of solvable S- arithmetic groups: an example” (to appear)] is considered; new proofs, using a version of the criterion, of some of their results are given. (The 4-by-4 case of this family of matrix groups is the well known example, due to Abels, of a finitely presented soluble group without the maximum condition for normal subgroups.)
Reviewer: J.R.J.Groves

##### MSC:
 20J05 Homological methods in group theory 20F05 Generators, relations, and presentations of groups 20F38 Other groups related to topology or analysis 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20E32 Simple groups 57M05 Fundamental group, presentations, free differential calculus 55U05 Abstract complexes in algebraic topology 55P99 Homotopy theory
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