Transitive bounded-degree 2-expanders from regular 2-expanders
Abstract
A two-dimensional simplicial complex is called d- regular if every edge of it is contained in exactly d distinct triangles. It is called ε-expanding if its up-down two-dimensional random walk has a normalized maximal eigenvalue which is at most 1-ε. In this work, we present a class of bounded degree 2-dimensional expanders, which is the result of a small 2-complex action on a vertex set. The resulted complexes are fully transitive, meaning the automorphism group acts transitively on their faces. Such two-dimensional expanders are rare! Known constructions of such bounded degree two-dimensional expander families are obtained from deep algebraic reasonings (e.g. coset geometries). We show that given a small d-regular two-dimensional ε-expander, there exists an ε'=ε'(ε) and a family of bounded degree two-dimensional simplicial complexes with a number of vertices goes to infinity, such that each complex in the family satisfies the following properties: * It is 4d-regular. * The link of each vertex in the complex is the same regular graph (up to isomorphism). * It is ε' expanding. * It is transitive. The family of expanders that we get is explicit if the one-skeleton of the small complex is a complete multipartite graph, and it is random in the case of (almost) general d-regular complex. For the randomized construction, we use results on expanding generators in a product of simple Lie groups. This construction is inspired by ideas that occur in the zig-zag product for graphs. It can be seen as a loose two-dimensional analog of the replacement product.