2016/07/26 by Lubotzky, Alexander, Luria, Zur, Rosenthal, Ron
#05E15 #05E18 #05E45 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1607.07734
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 [PR12] 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.