On groups and simplicial complexes

The theory of k-regular graphs is closely related to group theory. Every k-regular, bipartite graph is a Schreier graph with respect to some group G, a set of generators S (depending only on k) and a subgroup H. The goal of this paper is to begin to develop such a framework for k-regular simplicial complexes of general dimension d. Our approach does not directly generalize the concept of a Schreier graph, but still presents an extensive family of k-regular simplicial complexes as quotients of one universal object: the k-regular d-dimensional arboreal complex, which is itself a simplicial complex originating in one specific group depending only on d and k. Along the way we answer a question from Parzanchevski and Rosenthal (2016) on the spectral gap of higher dimensional Laplacians and prove a high dimensional analogue of Leighton's graph covering theorem. This approach also suggests a random model for k-regular d-dimensional multicomplexes.