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On the zeroeth complete cohomology of certain polycyclic groups. (English) Zbl 1227.20045

The paper under review offers an explicit projective resolution for a class of polycyclic groups and further provides a new counterexample to a conjecture of Brown stating that the least common multiple of the orders of the finite subgroups of a group \(G\) of finite virtual cohomological dimension annihilates the complete cohomology of \(G\). The first counterexample to this conjecture was found by A. Adem, [Bull. Lond. Math. Soc. 21, No. 6, 585-590 (1989; Zbl 0648.55017)].
Let \(N\) be the (multiplicative) group with generators \(a_1,a_2,\dots,a_n\), \(d\), subject to the relations: \[ [a_i,a_j]=1,\quad [a_i,d]=d^s=1, \] where \(i=1,2,\dots,n\), and \(j=1,2,\dots,n\). Then \(N\) is the direct product of a free Abelian group and a cyclic group of order \(s\). There is a well-known free resolution of \(N\) stated in the paper. Now, for any automorphism \(\tau\) of \(N\), let \(G\) denote the semidirect product of \(N\) by the infinite cyclic group \(\langle t\rangle\). By applying the construction \(\mathbb Z[G]\otimes_N-\) to the free resolution of \(N\), a resolution of \(\mathbb Z[G/N]\simeq\mathbb Z[\langle t\rangle]\) is obtained as \(\mathbb Z[G]\)-modules. The short exact sequence of \(\mathbb Z[G]\)-modules \[ 0\to\mathbb Z[G/N]@>\sigma>>\mathbb Z[G/N]@>\pi>>\mathbb Z\to 0 \] is used to coble together a projective resolution of \(G\). Above, \(\sigma(gN)=g(t-1)N\) and \(\pi(gN)=1\) for all \(g\in G\).
Let \(\widehat H^i(G;-)\), \(i\in\mathbb Z\), denote complete cohomology [D. J. Benson and J. F. Carlson, J. Pure Appl. Algebra 82, No. 2, 107-129 (1992; Zbl 0807.20044)]. For \(G\) the semidirect product above, it is proven that \(\widehat H^0(G;\mathbb Z)\) equals \(s\) if and only if \(\langle d\rangle\) has a complement in \(G\). A group \(G\) for which the subgroup \(\langle d\rangle\) does not admit a complement is a counterexample to Brown’s conjecture. One group providing such a counterexample is: \[ \langle a,d,t\mid [a,d]=d^5=1,\;a^t=a^{-1}d,\;d^t=d^4\rangle. \]

MSC:

20J06 Cohomology of groups
20F16 Solvable groups, supersolvable groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
18G05 Projectives and injectives (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
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References:

[1] DOI: 10.1112/blms/21.6.585 · Zbl 0648.55017
[2] DOI: 10.1016/0022-4049(90)90081-R · Zbl 0822.20053
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