Le but de cette note est de tenter d’expliquer les liens étroits qui unissent la théorie des empilements de cercles et des modules combinatoires et de comparer les approches à la conjecture de J.W. Cannon qui en découlent.
We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of Sp(4), which in addition satisfies the following property: for any n ≥ 3, there is in this family a walk associated with a reflection group of order 2n. Moreover, the case n = 4 corresponds to a process which appears naturally by studying quantum random walks on the dual of Sp(4). For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths...
Given an origami (square-tiled surface) with automorphism group , we compute the decomposition of the first homology group of into isotypic -submodules. Through the action of the affine group of on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.