Asphericity and Zero Divisors in Group Algebras by Ian J. Leary

Abstract S. V. Ivanov recently discovered a surprising connection between asphericity of certain spaces and the existence of zero-divisors in group algebras. We give a different proof of a more general result.

A recent paper of S. V. Ivanov exhibited a surprising link between the question of whether certain 2-complexes are aspherical and the question of whether the integral group ring of a torsion-free group can contain zero divisors [I]. Ivanov’s proof used a characterization of aspherical 2-complexes due to A. Yu. Ol’shanskii [O]. While studying Ivanov’s proof we realised that the following more general result could be proved using homology. Recall that a group is said to be acyclic if its Eilenberg-Mac Lane space has the same integral homology as a point.

Theorem 1. Let Y be an Eilenberg-Mac Lane space for a group H, build X by adding one 1-cell and one 2-cell to Y, and let G = π_{1}(X). Suppose that the kernel of the induced map from H to G is acyclic, and that H_{1}(X, Y) is finite. If X is not an Eilenberg-Mac Lane space for G, then there are zero divisors in the integral group ring ℤG. Moreover, there is a zero divisor in ℤG whose support in G is bounded by the number of times that the new 1-cell occurs (with either orientation) within the boundary of the new 2-cell.

Remarks. (1). The main theorem of [I] is equivalent to Theorem 1 under the additional hypotheses that H should admit a 2-dimensional Eilenberg-Mac Lane space and that the map from H to G should be injective. (2). The main case of interest is the case when G is torsion-free, as it is a notorious open problem whether ℤG can contain zero divisors in this case.

One of the zero divisors constructed may be described explicitly, as in [I]. Let Z be the intermediate space made by attaching the 1-cell but not the 2-cell to Y. Then the fundamental group of Z is the free product of H and an infinite cyclic group, with generator x say. The attaching map for the 2-cell may be described up to homotopy by a word

h_{1}x^{ϵ_{1}}h_{2} ⋯ h_{n}x^{ϵ_{n}},

where h_{i} ∈ H and ϵ_{i} = ±1. One of the zero divisors is the Fox derivative or ‘free derivative’ of this word with respect to x , i.e., the element a ∈ ℤG defined as

a = ∑_{i=1}^{n} ϵ_{i}h_{1} ⋯ h_{i}x^{(ϵ_{i}−1)/2}.

For more information concerning free derivatives, see for example [B, p45 and p90].

Note that the support of the element a has size at most n, and the image of a under the augmentation map ϵ : ℤG → ℤ is given by

ϵ(a) = ∑_{i=1}^{n} ϵ_{i}.

Proof of Theorem 1. Let f : H → G be the homomorphism induced by the inclusion of Y in X. Let X̃ denote the universal cover of X, and let Ỹ denote the subcomplex of X̃ consisting of lifts of cells of Y. The connected components of Ỹ are in bijective correspondence with the cosets of f(H) in G, and each component of Ỹ is an Eilenberg-Mac Lane space for a group isomorphic to ker(f). By hypothesis, ker(f) is acyclic, and hence H_{i}(Ỹ) = 0 for i > 0.

On the other hand, X̃ is connected and simply-connected, but is not contractible, and so by Hurewicz’s theorem there exists i > 1 for which H_{i}(X) ≠ {0}.

Since X consists of Y together with one 1-cell and one 2-cell, X̃ consists of Ỹ together with one free G-orbit of 1-cells and one free G-orbit of 2-cells. The relative cellular chain complex C_{∗}(X̃, Ỹ) is therefore

0 ⟶ ℤG⟶^{d}ℤG ⟶ 0,

where the non-trivial groups are in degrees 1 and 2, and the differential d corresponds to multiplication (on the right) by the element a described above [B, p45].

The long exact sequence in homology for the pair (X̃, Ỹ) shows that H_{i}(X̃) = 0 for i > 2, and that

H_{2}(X̃) ≅ H_{2}(X̃, Ỹ).

It follows that H_{2}(X̃, Ỹ) ≠ {0}, and hence that multiplication by a in ℤG has non-trivial kernel. It remains to show that a ≠ 0. The relative cellular chain complex C_{∗}(X, Y) is

0 ⟶ ℤ⟶^{d′}ℤ ⟶ 0,

where the non-trivial groups are in degrees 1 and 2, and the differential d′ corresponds to multiplication by ϵ(a). Hence the hypothesis that H_{1}(X, Y) be finite implies that ϵ(a) ≠ 0 and hence a ≠ 0.

Remarks. (3). Note that the hypothesis that H_{1}(X, Y) be finite is only used to show that a ≠ 0. Without this hypothesis, the same argument shows that either a = 0 or a is a zero divisor. (4). When we showed this argument to Noel Brady, he pointed out that it also applies to the case when non-aspherical X is made by attaching any number of 1-cells and one 2-cell to aspherical Y.

REFERENCES

[B] K. S. Brown, “Cohomology of Groups”, Graduate Texts in Mathematics 87, Springer (1982).

[I] S. V. Ivanov, “An Asphericity Conjecture and Kaplansky Problem on Zero Divisors”, J. Algebra 216 (1999), 13–19.

[O] A. Yu. Ol’shanskii, “Geometry of Defining Relations in Groups”, Nauka, Moscow, 1989; English translation in Math. Appl. (Soviet Ser.), Vol. 70, Kluwer Academic, Dordrecht, 1991.

Author’s address:

Faculty of Mathematical Studies,

University of Southampton, Southampton, SO17 1BJ.

email: ijl@maths.soton.ac.uk