Call diffeomorphisms satisfying (2) Artin-Mazur (A-M) diffeomorphisms. In [I.1] we give an elementary proof of an extension of Artin–Mazur’s result, namely, we prove that A-M diffeomorphisms with only hyperbolic periodic points are dense in Diffr(M). In [I.2] we construct Baire generic diffeomorphisms inside of an open set (a Newhouse domain) in Diffr(M) of with superexponential (even an arbitr...

We observe that, for fixed n ≥ 3, each of the Artin groups of finite type An, Bn = Cn, and affine type Ãn−1 and C̃n−1 is a central extension of a finite index subgroup of the mapping class group of the (n + 2)-punctured sphere. (The centre is trivial in the affine case and infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz on abstract commensurators of surface mapping...

In both algebraic geometry and coding theory, there is a great deal of interest in finding curves with many rational points. In particular, the correspondence between trace codes and Artin-Schreier curves gives a relation between the weights of codewords and the number of rational points on such curves, low weight codewords yielding curves with a large number of rational points. Further, subcod...

We consider selfinjective Artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators. For such an algebra, we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules. This provides a unified approach to computing lower bounds for the representation dimension of group algebras, exterior alge...

We begin the paper with a simple formula for the second integral homology of a range of Artin groups. The formula is derived from a polytopal classifying space. We then introduce the notion of a twisted Artin group and obtain polytopal classifying spaces for a range of such groups. We demonstrate that these explicitly constructed spaces can be implemented on a computer and used in homological c...

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = 〈a, b | prod(a, b; k) = prod(b, a; k)〉 where prod(s, t; k) = ststs..., with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generators {a, b, a, b−1} are rational. We provide explicit formulas for the series.