Topology and Geometry of Random 2-Dimensional Hypertrees

Abstract

A hypertree, or \(\mathbb {Q}\)-acyclic complex, is a higher-dimensional analogue of a tree. We study random 2-dimensional hypertrees according to the determinantal measure suggested by Lyons. We are especially interested in their topological and geometric properties. We show that with high probability, a random 2-dimensional hypertree T is aspherical, i.e., that it has a contractible universal cover. We also show that with high probability the fundamental group \(\pi _1(T)\) is hyperbolic and has cohomological dimension 2.