An elementary approach to dessins d'enfants and the Grothendieck-Teichm\"uller group

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field extensions, and some classes of algebraic curves), some of which are naturally endowed with an action of the absolute Galois group of the rational field. We prove that the action is faithful. Eventually we prove that this absolute Galois group embeds into the Grothendieck-Teichm\"uller group $GT_0$ introduced by Drinfel'd. There are explicit approximations of $GT_0$ by finite groups, and we hope to encourage computations in this area. Our treatment includes a result which has not appeared in the literature yet: the Galois action on the subset of regular dessins - that is, those exhibiting maximal symmetry -- is also faithful.Comment: 58 pages, about 30 figures. Corrected a few typos. This version should match the published paper in L'enseignement Mathematiqu

Examples of Sweedler cohomology in positive characteristic

In this paper we provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (Z/2)^r, in all degrees, over a field of characteristic 2. The result is strikingly different from the characteristic 0 analog. Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group.Comment: With this update we pretty much revert to version 2 (including the old title), for the good reason that this version has been accepted for publication in Comm. Alg. (without part II which appeared in version 3). So this final paper is identical with the published on

Chow rings and cobordism of some Chevalley groups

We compute the cobordism ring $MU^*(BG)\otimes \mathbb{F}_p$ when $G$ is a Chevalley group. In the particular case of the general linear group, we prove that it agrees with the Chow ring (as defined by Totaro).Comment: 18 pages, uses psp.cls, to appear shortly in Math. Proc. Cam. Phil. So

Cayley graphs and automatic sequences

International audienceWe study those automatic sequences which are produced by an automaton whose underlying graph is the Cayley graph of a finite group. For 2-automatic sequences, we find a characterization in terms of what we call homogeneity, and among homogeneous sequences, we single out those enjoying what we call self-similarity. It turns out that self-similar 2-automatic sequences (viewed up to a permutation of their alphabet) are in bijection with many interesting objects, for example dessins d'enfants (covers of the Riemann sphere with three points removed). For any p we show that, in the case of an automatic sequence produced "by a Cayley graph", the group and indeed the automaton can be recovered canonically from the sequence. Further, we show that a rational fraction may be associated to any automatic sequence. To compute this fraction explicitly, knowledge of a certain graph is required. We prove that for the sequences studied in the first part, the graph is simply the Cayley graph that we start from, and so calculations are possible. We give applications to the study of the frequencies of letters