Finiteness properties of groups.

*(English)*Zbl 0613.20033This 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.)

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 |

##### Keywords:

homology; finiteness conditions; type \(FP_ n\); finite presentation; CW- complex; acyclic; n-good complex; finitely presented infinite simple groups; groups of homeomorphisms
Full Text:
DOI

##### References:

[1] | Abels, H., An example of a finitely presented solvable group, (), 205-211 |

[2] | H. Abels and K.S. Brown, Finiteness properties of solvable S-arithmetic groups: An example, J. Pure Appl. Algebra, in this volume. · Zbl 0617.20020 |

[3] | Åberg, H., Bieri-Strebel valuations (of finite rank), Proc. London math. soc., 52, 3, 269-304, (1986) · Zbl 0588.20026 |

[4] | Bieri, R., Homological dimension of discrete groups, (), London · Zbl 0357.20027 |

[5] | Bieri, R., A connection between the integral homology and the centre of a rational linear group, Math. Z., 170, 263-266, (1980) · Zbl 0427.20042 |

[6] | Bieri, R.; Eckmann, B., Finiteness properties of duality groups, Comment. math. helv., 49, 74-83, (1974) · Zbl 0279.20041 |

[7] | Brown, K.S., Complete Euler characteristics and fixed-point theory, J. pure appl. algebra, 24, 103-121, (1982) · Zbl 0493.20033 |

[8] | Brown, K.S., Cohomology of groups, (1982), Springer Berlin · Zbl 0367.18012 |

[9] | Brown, K.S., Presentations for groups acting on simply-connected complexes, J. pure appl. algebra, 32, 1-10, (1984) · Zbl 0545.20022 |

[10] | Brown, K.S.; Geoghegan, R., An infinite-dimensional torsion-free FP_∞ group, Invent. math., 77, 367-381, (1984) · Zbl 0557.55009 |

[11] | Dydak, J., A simple proof that pointed connected FANR-spaces are regular fundamental retracts of ANR’s, Bull. polon. acad. sci. ser. math. astronom. phys., 25, 55-62, (1977) · Zbl 0357.55018 |

[12] | Dydak, J., 1-movable continua need not be pointed 1-movable, Bull. polon. acad. sci. ser. sci. math. astronom. phys., 25, 485-488, (1977) · Zbl 0361.54019 |

[13] | Epstein, D.B.A., The simplicity of certain groups of homeomorphisms, Compositio math., 22, 165-173, (1970) · Zbl 0205.28201 |

[14] | Folkman, J., The homology groups of a lattice, J. math. mech., 15, 631-636, (1966) · Zbl 0146.01602 |

[15] | Freyd, P.; Heller, A., Splitting homotopy idempotents, II, (1979), unpublished manuscript · Zbl 0786.55008 |

[16] | Grayson, D., Finite generation of K-groups of a curve over a finite field, (), 69-90, [after D. Quillen] |

[17] | Higman, G., Finitely presented infinite simple groups, () · Zbl 0104.02101 |

[18] | Houghton, C.H., The first cohomology of a group with permutation module coefficients, Arch. math., 31, 254-258, (1978/1979) · Zbl 0377.20044 |

[19] | Jónsson, B.; Tarski, A., On two properties of free algebras, Math. scand., 9, 95-101, (1961) · Zbl 0111.02002 |

[20] | McKenzie, R.; Thompson, R.J., An elementary construction of unsolvable word problems in group theory, (), 457-478 |

[21] | Quillen, D., Finite generation of the groups Ki of rings of algebraic integers, (), 179-198 |

[22] | Scott, P., Ends of pairs of groups, J. pure appl. algebra, 11, 179-198, (1977) · Zbl 0368.20021 |

[23] | Serre, J.-P., Cohomologie des groupes discrets, Ann. of math. studies, 70, 77-169, (1971) · Zbl 0273.57022 |

[24] | Thompson, R.J., Embeddings into finitely generated simple groups which preserve the word problem, (), 401-441 |

[25] | Vogtmann, K., Spherical posets and homology stability for On,n, Topology, 20, 119-132, (1981) · Zbl 0455.20031 |

[26] | Wiegold, J., Transitive groups with fixed-point free permutations II, Arch. math., 29, 571-573, (1977) · Zbl 0382.20029 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.