On groups and simplicial complexes. (May 2018)

• Record Type: Journal Article
• Authors: Lubotzky, Alexander; Luria, Zur; Rosenthal, Ron
• Is Part Of: European journal of combinatorics. Volume 70(2018)
• Page Start: 408
• Page End: 444
• Publication Date: 2018-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2018.01.009
• Languages: English
• ISSNs: 0195-6698

Abstract

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 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.