Ralph Stöhr

Geometry and cohomology in group theory

1. On the cohomology of SL2(Z[1/p]) A. Adem and N. Naffah 2. Cohomology of sporadic groups, finite loop spaces and the Dickson Invariants D. J. Benson 3. Kernels of actions on non-positively curved spaces R. Bieri and R. Geoghegan 4. Cyclic groups acting on free Lie algebras R. M. Bryant 5. Cohomology, representations and quotient categories of modules J. F. Carlson 6. Protrees and L-trees I. M. Chiswell 7. Homological techniques for strongly graded rings: a survey J. Cornick 8. Buildings are CAT(0) M. W. Davis 9. On subgroups of Coxeter groups W. Dicks and I. J. Leary 10. The p-primary Farrell cohomology of Out(Fp-1) H. H. Glover, G. Mislin and S. N. Voon 11. On Tychonoff groups R. I. Grigorchuk 12. Word growth of Coxeter groups D. L. Johnson 13. Poly-surface groups F. E. A. Johnson 14. Analytic versions of the zero divisor conjecture P. A. Linnell 15. On the geometric invariants of soluble groups of finite Prufer rank H. Meinert 16. Some constructions relating to hyperbolic groups K. V. Mikhajlovskii and A. Yu. Olshanskii 17. Free actions of Abelian groups on groups P. M. Neumann and P. J. Rowley 18. Finitely presented soluble groups J. S. Wilson.

On the module structure of free Lie algebras

We study the free Lie algebra L over a field of non-zero characteristic p as a module for the cyclic group of order p acting on L by cyclically permuting the elements of a free generating set. Our main result is a complete decomposition of L as a direct sum of indecomposable modules.

THE EQUATION [x,u] + [y,v] = 0 IN FREE LIE ALGEBRAS

We investigate equations of the form [x,u] + [y,v] = 0 over a free Lie algebra L. In the case where u and v are free generators of L, we exhibit two series of solutions, we work out the dimensions of the homogeneous components of the solution space, and we determine its radical. In the general case we show that the results on free generator coefficients are sufficient to obtain the solution space up to finite codimension. As an application we determine the radical of the bilinear equation [x1,x2] + [x3,x4] = 0.

Homology of free Lie powers and torsion in groups

LetG be a group that is given by a free presentationG=F/R, and letγ4R denote the fourth term of the lower central series of R. We show that ifG has no elements of order 2, then the torsion subgroup of the free central extensionF/[γ4R,F] can be identified with the homology groupRγ6(G, ℤ/2ℤ). This is a consequence of our main result which refers to the homology ofG with coefficients in Lie powers of relation modules.

On algebraic sets over metabelian groups

Abstract We investigate algebraic sets over certain finitely generated torsion-free metabelian groups. The class of groups under consideration is the class of so-called ρ -groups. It consists of all wreath products of finitely generated free abelian groups and their subgroups. In particular, it includes all free metabelian groups of finite rank. Our main result is a characterization of certain irreducible algebraic sets over ρ -groups. More precisely, we consider irreducible algebraic sets which are determined by a system of equations in n indeterminates. For their coordinate groups, we introduce a discrete invariant called the relative characteristic. This is an ordered pair of non-negative integers. We determine the structure of the coordinate group of the n -dimensional affine space, and show that its relative characteristic is (n, n ). Then we characterize the irreducible algebraic sets of relative characteristic (n, n ) and (0, k ) where 0 ≤ k ≤ n . We also obtain some examples of somewhat unusual algebraic sets over ρ -groups, thus demonstrating that algebraic sets over these groups are much more varied and complicated than, say, algebraic sets over free groups.

Subalgebras of free restricted Lie Algebras

A theorem independently due to A.I. Shirshov and E. Witt asserts that every subalgebra of a free Lie algebra (over a field) is free. The main step in Shirshovs proof is a little known but rather remarkable result: if a set of homogeneous elements in a free Lie algebra has the property that no element of it is contained in the subalgebra generated by the other elements, then this subset is a free generating set for the subalgebra it generates. Witt also proved that every subalgebra of a free restricted Lie algebra is free. Later G.P. Kukin gave a proof of this theorem in which he adapted Shirshovs argument. The main step is similar, but it has come to light that its proof contains substantial gaps. Here we give a corrected proof of this main step in order to justify its applications elsewhere.

Lie powers of modules for GL(2,p)

In a previous paper [12] with almost the same title, two of the authors considered the Lie powersL(V ) of the natural moduleV for GL(2,p) (wherep is prime). For the case whenn is not divisible byp, it was shown there that the indecomposable direct summands of L(V ) are either simple or projective, and two methods were described for calculating the relevant Krull–Schmidt multiplicities. The hard part of the paper dealt with the case of n divisible byp, but only under the assumption that p 3: for p 5, this case was left open. The purpose of the present paper is to complete this investigation. For background and motivation, see [12]. The main qualitative result (see Corollary 2.2) is rather more general:if V is any finite direct sum of simple modules and projective indecomposable modules for GL(2,p) over a field of characteristic p, then the nonsimple, nonprojective indecomposable direct summands in any Lie power of V can only be of dimension p − 1 and composition length 2. The proof is constructive, and most of it is of even wider application: it amounts to a method for calculating Krull–Schmidt multiplicities in Lie powers of arbitrary finite-dimensional GL(2,p)-modules in characteristic p. (We do not think that this method improves computation in the cases already covered in [12].) One may well question how far it is practical to perform the actual calculations, even if one only wants Lie powers of the natural module, but the qualitative result paraphrased above demonstrates the penetrating power of this approach. Remarkably, the claim about the composition length of the nonsimple, nonprojective indecomposables which occur is equally true for the Lie powers of the natural module

Free central extensions of groups and modular Lie powers of relation modules

The most prominent special case of our main result is that the free centre-by-(nilpotent of class ())-by-abelian groups are torsion-free whenever is divisible by at least two distinct primes. This is in stark contrast to the case where is a prime or , where these relatively free groups contain non-trivial elements of finite order.

On torsion in certain free centre-by-soluble groups

Abstract We study the homology of iterated exterior squares of relation modules to obtain information about torsion in certain free central extensions of groups. In particular, we detect elements of finite order in free centre-by-soluble of length n+1 groups with nilpotent of class 2nn commutator subgroup, and give a complete description of the torsion subgroup in the case n=2.

Torsion in free center-by-nilpotent-by-abelian Lie rings of rank 2

ABSTRACT For c≥2, the free center-by-(nilpotent-of-class-c-1)-by-abelian Lie ring on a set X is the quotient where L is the free Lie ring on X, and denotes the cth term of the lower central series of the derived ideal of L. In this paper, we give a complete description of the torsion subgroup of its additive group in the case where |X| = 2 and c is a prime number.