Garland's Technique for Posets and High Dimensional Grassmannian Expanders

Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [11] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [2,14], an equivalence with a global random walk definition, generalizing [6] and a trickling down theorem, generalizing [20]. In particular, we show that some posets, such as the Grassmannian poset, exhibit qualitatively stronger trickling down effect than simplicial complexes. Using these methods, and the novel idea of posetification to Ramanujan complexes [18,19], we construct a constant degree expanding Grassmannian poset, and analyze its expansion. This it the first construction of such object, whose existence was conjectured in [6].