The stable braid group and the determinant of the Burau representation

This article gives certain fibre bundles associated to the braid groups which are obtained from a translation as well as conjugation on the complex plane. The local coefficient systems on the level of homology for these bundles are given in terms of the determinant of the Burau representation. De Concini, Procesi, and Salvetti [Topology 40 (2001) 739--751] considered the cohomology of the n-th braid group B_n with local coefficients obtained from the determinant of the Burau representation, H^*(B_n;Q[t^{+/-1}]). They show that these cohomology groups are given in terms of cyclotomic fields. This article gives the homology of the stable braid group with local coefficients obtained from the determinant of the Burau representation. The main result is an isomorphism H_*(B_infty; F[t^{+/-1}])-->H_*(Omega^2S^3; F) for any field F where Omega^2S^3 denotes the double loop space of the 3-connected cover of the 3-sphere. The methods are to translate the structure of H_*(B_n;F[t^{+/-1}]) to one concerning the structure of the homology of certain function spaces where the answer is computed.Comment: This is the version published by Geometry & Topology Monographs on 29 January 200

The square root law and structure of finite rings

Let $R$ be a finite ring and define the hyperbola $H=\{(x,y) \in R \times R: xy=1 \}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following "square root law" bound holds with a constant $C>0$ for all non-trivial characters $\chi$ on $R^2$: $$\left| \sum_{(x,y)\in H}\chi(x,y)\right|\leq C\sqrt{|H|}.$$ Then, with a finite number of exceptions, those rings are fields. For rings of even order we show that there are other infinite families given by Boolean rings and Boolean twists which satisfy this square-root law behavior. We classify the extremal rings, those for which the left hand side of the expression above satisfies the worst possible estimate. We also describe applications of our results to problems in graph theory and geometric combinatorics. These results provide a quantitative connection between the square root law in number theory, Salem sets, Kloosterman sums, geometric combinatorics, and the arithmetic structure of the underlying rings

Basis-conjugating automorphisms of a free group and associated Lie algebras

Let F_n = denote the free group with generators {x_1,...,x_n}. Nielsen and Magnus described generators for the kernel of the canonical epimorphism from the automorphism group of F_n to the general linear group over the integers. In particular among them are the automorphisms chi_{k,i} which conjugate the generator x_k by the generator x_i leaving the x_j fixed for j not k. A computation of the cohomology ring as well as the Lie algebra obtained from the descending central series of the group generated by chi_{k,i} for i<k is given here. Partial results are obtained for the group generated by all chi_{k,i}.Comment: This is the version published by Geometry & Topology Monographs on 22 February 200

The essential ideal in group cohomology does not square to zero

Let G be the Sylow 2-subgroup of the unitary group $SU_3(4)$. We find two essential classes in the mod-2 cohomology ring of G whose product is nonzero. In fact, the product is the ``last survivor'' of Benson-Carlson duality. Recent work of Pakianathan and Yalcin then implies a result about connected graphs with an action of G. Also, there exist essential classes which cannot be written as sums of transfers from proper subgroups. This phenomenon was first observed on the computer. The argument given here uses the elegant calculation by J. Clark, with minor corrections.Comment: 9 pages; three typos corrected, one was particularly confusin