Volume 2008, Issue 619

We show that every automorphism α of a free group F k of finite rank k has asymptotically periodic dynamics on F k and its boundary ∂ F k : there exists a positive power α q such that every element of the compactum converges to a fixed point under iteration of α q . Further results about the dynamics of α, as well as an extension from F k to word-hyperbolic groups, are given in the later sections.

Using Frobenius-Stickelberger-type relations for hyperelliptic curves (Y. Ônishi, Proc. Edinb. Math. Soc. (2) 48 (2005), 705–742), we provide certain addition formulae for any symmetric power of such curves, which hold on the strata W k , the pre-images in the Jacobian of the classical Wirtinger varieties. In an appendix, we give similar relations for a trigonal curve y 3 = ( x – b 1 )( x – b 2 )( x – b 3 )( x – b 4 ).

We provide new evidence towards Eisenbud-Harris conjecture in Castelnuovo theory concerning algebraic curves of high genus in P n .

We show how to make the additive Chow groups of Bloch-Esnault, Rülling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety. In case the base field k admits resolution of singularities, these properties allow us to apply the technique of Guillén and Navarro Aznar to define the additive Chow groups “with log poles at infinity” for an arbitrary finite-type k -scheme X . This theory has all the usual properties of a Borel-Moore theory on finite type k -schemes: it is covariantly functorial for projective morphisms, contravariantly functorial for morphisms of smooth schemes, and has a projective bundle formula, homotopy property, and Mayer-Vietoris and localization sequences. Finally, we show that the regulator map defined by Park from the additive Chow groups of 1-cycles to the modules of absolute Kähler differentials of an algebraically closed field of characteristic zero is surjective, giving evidence of a conjectured isomorphism between these two groups.

This paper develops an abstract framework for constructing “seminormal forms” for cellular algebras. That is, given a cellular R -algebra A which is equipped with a family of JM- elements we give a general technique for constructing orthogonal bases for A , and for all of its irreducible representations, when the JM-elements separate A . The seminormal forms for A are defined over the field of fractions of R . Significantly, we show that the Gram determinant of each irreducible A -module is equal to a product of certain structure constants coming from the seminormal basis of A . In the non-separated case we use our seminormal forms to give an explicit basis for a block decomposition of A .

Hypertoric varieties are determined by hyperplane arrangements. In this paper, we use stacky hyperplane arrangements to define the notion of hypertoric Deligne-Mumford stacks. Their orbifold Chow rings are computed. As an application, some examples related to crepant resolutions are discussed.

For each submanifold of a stratified group, we find a number and a measure only depending on its tangent bundle, the grading and the fixed Riemannian metric. In two step stratified groups, we show that such number and measure coincide with the Hausdorff dimension and with the spherical Hausdorff measure of the submanifold with respect to the Carnot-Carathéodory distance, respectively. Our main technical tool is an intrinsic blow-up at points of maximum degree. We also show that the intrinsic tangent cone to the submanifold at these points is always a subgroup. Finally, by direct computations in the Engel group, we show how our results can be extended to higher step stratified groups, provided the submanifold is sufficiently regular.

The result in our article “The Chow rings of G 2 and Spin(7)” depends on a computation by Yagita, who in turn ascribes a critical contribution, concerning the Chow ring of Spin(7), to Totaro. However in a private communication, Totaro has pointed out to me that he has not published the result anywhere. At the same time, he also explained that a recent paper of Yagita and Schuster gives an optimal counterexample to a conjecture of his on Chow rings—modulo the same result without a published proof. In this addendum, we give a direct proof for this missing ingredient. Using the notations and background of our aforementioned article, it is rather straightforward, and thus the proof fits well as a concluding point. Then we present the relationship with the article of Yagita and Schuster, and with Totaro's conjecture.