Bérénice Delcroix-Oger: Homology of the hypertree poset

Abstract

The hypertree poset has been introduced by McCullough and Miller in 1996 in the frame of geometric group theory. McCullough and Miller, soon followed by McCammond, Meier, Brady and Jensen used the hypertree poset to study the cohomology of some automorphism groups of free groups and free products. McCammond and Meier proved that hypertree posets are Cohen-Macaulay and computed the Moebius number of the poset of hypertrees on n vertices : (n-1)^{n-2}. This number is also the dimension of a vector space linked with PreLie operad. We answer here to a conjecture of Chapoton by linking the action of the symmetric group on the unique homology group of the hypertree poset with PreLie operad. This study uses the notion of species of structure, which will be recalled as well as the notion of operad.