##
**Homology of free abelian extensions of groups.**
*(English.
Russian original)*
Zbl 0798.20043

Math. USSR, Sb. 72, No. 2, 503-518 (1992); translation from Mat. Sb. 182, No. 4, 526-542 (1991).

Let \(G\) be a group presented as a factor group \(G = F/N\) of a free group \(F\). Let \(\Phi = F/N'\), which is known as a free abelian extension of \(G\). Ground breaking work on the homology groups of \(\Phi\) with trivial integer coefficients was done by the second author [Commun. Algebra 16, No. 12, 2447-2533 (1988; Zbl 0682.20038)]. In particular he obtained a great deal of information about the odd periodic parts of these homology groups. In this beautiful paper this work is carried much further by obtaining, for each odd prime \(p\), an explicit expression for the \(p\)- component of \(H_ n(G,Z)\) (\(n \geq 2\)) as a direct sum of homology groups of \(G\) in various dimensions with coefficients in \(Z_ p\), the integers \(\text{mod }p\). The expression is valid for groups with no \(p\)- torsion. It follows of course in particular that this \(p\)-component is of exponent dividing \(p\). This had previously been conjectured by the second author.

To explain these results in more detail, let \(M = N/N'\) be the relation module coming from the above presentation. Thus, we have the exact sequence \(1 \to M \to \Phi \to G \to 1\). Let \(tA\) denote the torsion subgroup of an abelian group \(A\), and \(t_ nA\) denote the set of elements of \(A\) annihilated by some power of the integer \(n\). Let \(\wedge^ n A\) denote the \(n\)-th exterior power of a \(G\)-module \(A\), viewed as a \(G\)-module via the diagonal action. By computing in the spectral sequence of the above extension, Kuz’min obtained an isomorphism \(tH_ n(\Phi,Z[{1\over 2}]) \cong t_ n(\wedge^ n M \otimes_ G Z[{1\over 2}])\), and an exact sequence \[ \begin{split} 0 \to t_{n-1} H_ 2(G,\wedge^{n - 1}M\otimes Z[\textstyle{1\over 2}])\to \wedge^ nM\otimes_ G Z[\textstyle{1\over 2}] \to\\ \to H_ n(\Phi,Z[\textstyle{1\over 2}])\to t_{n - 1} H_ 1(G,\wedge^{n - 1} M \otimes Z[\textstyle{1\over 2}]) \to 0.\end{split} \] Let \(T_ n\) be the periodic part of \(H_ n(\Phi,Z[{1\over 2}])\), and let \(C_ n\) and \(K_ n\) denote the cokernel and kernel, respectively, of the map \(\wedge^ n M\otimes_ G Z[{1\over 2}] \to H_ n(\Phi,Z[{1\over 2}])\) induced by the inclusion \(M \to \Phi\). This is the middle map in the exact sequence above.

The main theorem obtains an exact description of \(T_ n\), \(C_ n\) and \(K_ n\) in terms of homology groups of \(G\), under certain hypotheses of torsion freeness. To describe this, the following notation is used. If \(A\) is an abelian group and \(m \geq 0\), then \(mA\) denotes the direct sum of \(m\) copies of \(A\). If \(f(x) = \sum m_ k x^ k\) is a polynomial with non-negative integer coefficients, and \(A\) is a \(G\)-module, then \(fH_ n(G,A) =\bigoplus_ k m_ k H_{n + k}(G,A)\). Next, polynomials \(f^ p_ n\), with non-negative integer coefficients are recursively defined for each prime \(p\) and integer \(n > 1\), as follows. We put \(f^ p_ n = 0\) unless \(n\) is congruent to 0 or \(1 \bmod p\) (the Russian version says if \(n\) is congruent to 0 or \(1 \bmod p\)). Then we put \(f^ p_ n = x^ 2\) if \(n = p\); \(f^ p_ n = x f^ p_{n - 1}\) if \(n \equiv 1\pmod p\), and \(f^ p_ n = x^ 2 f^ p_{n - p} + f^ p_{n/p}\) if \(n \equiv 0\pmod p\) but \(n > p\). Then the main theorem is as follows.

Theorem. Let \(p\) be an odd prime, and let \(G\) be a group without \(p\)- torsion. If \(n \equiv 0 \pmod p\), then \(t_ p T_ n \cong f^ p_ n H_ n(G,Z_ p)\), and if \(n \equiv1\pmod p\), and \(n > 1\), then \(t_ p C_ n \cong f^ p_{n - 1} H_ n(G,Z_ p)\), while \(t_ pK_ n \cong f^ p_{n - 1}H_{n+1}(G,Z_ p)\). Clearly, if \(G\) has no \(n\)- torsion, then \(T_ n\) itself will be the direct sum of the \(p\)- components given by the theorem, and so will have exponent dividing the product of the distinct prime divisors of \(n\).

According to the isomorphism and exact sequence given at the beginning, the proof of the theorem reduces to questions about the homology of \(G\) with coefficients in exterior powers of the relation module \(M\). This is handled by transforming it to a question about homology with coefficients in the symmetric powers of the augmentation ideal \(\Delta\) of the group ring of \(G\) with coefficients in the ring of integers localized at \(p\). Namely, there is an isomorphism \(t_ p H_ k(G,\wedge^ n M)\cong t_ p H_{k + n}(G,\Delta^ n)\), where \(\Delta^ n\) is the \(n\)-th symmetric power of \(\Delta\). This is valid for all primes including 2. The bulk of the work then is devoted to calculating the homology of \(G\) with coefficients in these symmetric powers.

Analogous results about cohomology are mentioned, and the last section gives some properties of these interesting polynomials \(f^ p_ n\).

To explain these results in more detail, let \(M = N/N'\) be the relation module coming from the above presentation. Thus, we have the exact sequence \(1 \to M \to \Phi \to G \to 1\). Let \(tA\) denote the torsion subgroup of an abelian group \(A\), and \(t_ nA\) denote the set of elements of \(A\) annihilated by some power of the integer \(n\). Let \(\wedge^ n A\) denote the \(n\)-th exterior power of a \(G\)-module \(A\), viewed as a \(G\)-module via the diagonal action. By computing in the spectral sequence of the above extension, Kuz’min obtained an isomorphism \(tH_ n(\Phi,Z[{1\over 2}]) \cong t_ n(\wedge^ n M \otimes_ G Z[{1\over 2}])\), and an exact sequence \[ \begin{split} 0 \to t_{n-1} H_ 2(G,\wedge^{n - 1}M\otimes Z[\textstyle{1\over 2}])\to \wedge^ nM\otimes_ G Z[\textstyle{1\over 2}] \to\\ \to H_ n(\Phi,Z[\textstyle{1\over 2}])\to t_{n - 1} H_ 1(G,\wedge^{n - 1} M \otimes Z[\textstyle{1\over 2}]) \to 0.\end{split} \] Let \(T_ n\) be the periodic part of \(H_ n(\Phi,Z[{1\over 2}])\), and let \(C_ n\) and \(K_ n\) denote the cokernel and kernel, respectively, of the map \(\wedge^ n M\otimes_ G Z[{1\over 2}] \to H_ n(\Phi,Z[{1\over 2}])\) induced by the inclusion \(M \to \Phi\). This is the middle map in the exact sequence above.

The main theorem obtains an exact description of \(T_ n\), \(C_ n\) and \(K_ n\) in terms of homology groups of \(G\), under certain hypotheses of torsion freeness. To describe this, the following notation is used. If \(A\) is an abelian group and \(m \geq 0\), then \(mA\) denotes the direct sum of \(m\) copies of \(A\). If \(f(x) = \sum m_ k x^ k\) is a polynomial with non-negative integer coefficients, and \(A\) is a \(G\)-module, then \(fH_ n(G,A) =\bigoplus_ k m_ k H_{n + k}(G,A)\). Next, polynomials \(f^ p_ n\), with non-negative integer coefficients are recursively defined for each prime \(p\) and integer \(n > 1\), as follows. We put \(f^ p_ n = 0\) unless \(n\) is congruent to 0 or \(1 \bmod p\) (the Russian version says if \(n\) is congruent to 0 or \(1 \bmod p\)). Then we put \(f^ p_ n = x^ 2\) if \(n = p\); \(f^ p_ n = x f^ p_{n - 1}\) if \(n \equiv 1\pmod p\), and \(f^ p_ n = x^ 2 f^ p_{n - p} + f^ p_{n/p}\) if \(n \equiv 0\pmod p\) but \(n > p\). Then the main theorem is as follows.

Theorem. Let \(p\) be an odd prime, and let \(G\) be a group without \(p\)- torsion. If \(n \equiv 0 \pmod p\), then \(t_ p T_ n \cong f^ p_ n H_ n(G,Z_ p)\), and if \(n \equiv1\pmod p\), and \(n > 1\), then \(t_ p C_ n \cong f^ p_{n - 1} H_ n(G,Z_ p)\), while \(t_ pK_ n \cong f^ p_{n - 1}H_{n+1}(G,Z_ p)\). Clearly, if \(G\) has no \(n\)- torsion, then \(T_ n\) itself will be the direct sum of the \(p\)- components given by the theorem, and so will have exponent dividing the product of the distinct prime divisors of \(n\).

According to the isomorphism and exact sequence given at the beginning, the proof of the theorem reduces to questions about the homology of \(G\) with coefficients in exterior powers of the relation module \(M\). This is handled by transforming it to a question about homology with coefficients in the symmetric powers of the augmentation ideal \(\Delta\) of the group ring of \(G\) with coefficients in the ring of integers localized at \(p\). Namely, there is an isomorphism \(t_ p H_ k(G,\wedge^ n M)\cong t_ p H_{k + n}(G,\Delta^ n)\), where \(\Delta^ n\) is the \(n\)-th symmetric power of \(\Delta\). This is valid for all primes including 2. The bulk of the work then is devoted to calculating the homology of \(G\) with coefficients in these symmetric powers.

Analogous results about cohomology are mentioned, and the last section gives some properties of these interesting polynomials \(f^ p_ n\).

Reviewer: B.Hartley (Manchester)

### MSC:

20J05 | Homological methods in group theory |

20F05 | Generators, relations, and presentations of groups |

20E22 | Extensions, wreath products, and other compositions of groups |

20E05 | Free nonabelian groups |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |