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Virtually soluble groups of type \(\text{FP}_\infty\). (English) Zbl 1276.20057

From the introduction: We prove that a virtually soluble group \(G\) of type \(\text{FP}_\infty\) admits a finitely dominated model for \(\underline EG\) of dimension the Hirsch length of \(G\). This implies in particular that the Brown conjecture is satisfied for virtually torsion-free elementary amenable groups.
In this note we show that a virtually soluble group of type VF, i.e., a virtually soluble group of type \(\text{FP}_\infty\) indeed admits a finitely dominated model for \(\underline EG\). In Section 2 we will show that such a group has only finitely many conjugacy classes of finite subgroups. Most of the remainder of this paper is then devoted to proving that centralizers of finite subgroups are also of type \(\text{FP}_\infty\). These centralizers are finitely presented [G. Baumslag and R. Bieri, Math. Z. 151, 249-257 (1976; Zbl 0356.20028)] and have finite index in the corresponding normalizers hence the conditions of Theorem 1.2 are satisfied. Furthermore, since a virtually soluble group of type \(\text{FP}_\infty\) also admits a finite dimensional model for \(\underline EG\), the claim that it admits a finitely dominated model for \(\underline EG\) now follows from Theorems 5.1 and 6.3 of W. Lück [J. Pure Appl. Algebra 149, No. 2, 177-203 (2000; Zbl 0955.55009)].
As nice applications of our theorem we show that for virtually torsion-free elementary amenable groups Brown’s conjecture is satisfied and that the property of admitting a finitely dominated \(\underline EG\) is a quasi-isometry invariant within the class of virtually soluble groups.
Also note that our main result can be extended to elementary amenable groups of type \(\text{FP}_\infty\). A result of J. A. Hillman and P. A. Linnell [J. Aust. Math. Soc., Ser. A 52, No. 2, 237-241 (1992; Zbl 0772.20010)] shows that elementary amenable groups of finite Hirsch length are locally finite-by-virtually soluble. In case these are of type \(\text{FP}_\infty\) they also have a bound on the orders of the finite subgroups [P. H. Kropholler, J. Pure Appl. Algebra 90, No. 1, 55-67 (1993; Zbl 0816.20042)] so we can reduce the above questions to questions on virtually soluble groups.

MSC:

20J05 Homological methods in group theory
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
20F19 Generalizations of solvable and nilpotent groups
43A07 Means on groups, semigroups, etc.; amenable groups
20E26 Residual properties and generalizations; residually finite groups
57S30 Discontinuous groups of transformations
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