On groups and simplicial complexes

Abstract

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 [PR16] 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.

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“…The Q-connectivity analysis allows for considering the connectedness of the system (Bozhenyuk and Ginis, 2013;Capitelli, 2016;Altmann, Bigdeli, Herzog, and Lu, 2016;Boissonnat and Mazauric, 2016;Seyed Fakhari, 2016;Chachólski, Scolamiero, and Vaccarino, 2017;Deo, 2018;Lubotzky, Luria, and Rosenthal, 2018;Luo and Tate, 2018) more deeply than traditional graph connectivity studies, and the presence of mutual influence of simplicial blocks of the system through a chain of connections between them (sets of vertices V and a family of nonempty subsets of these verticessimplexes is established δ). The sets of vertices and their corresponding simplexes form simplicial complexes (Wheeler, 2017).…”