Couboundary Expansion of Sheaves on Graphs and Weighted Mixing Lemmas

08/02/2022

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by   Uriya A. First, et al.

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We study the coboundary expansion of graphs, but instead of using 𝔽_2 as the coefficient group when forming the cohomology, we use a sheaf on the graph. We prove that if the graph under discussion is a good expander, then it is also a good coboundary expander relative to any constant augmented sheaf (equivalently, relative to any coefficient group R); this, however, may fail for locally constant sheaves. We moreover show that if we take the quotient of a constant augmented sheaf on an excellent expander graph by a "small" subsheaf, then the quotient sheaf is still a good coboundary expander. Along the way, we prove a new version of the Expander Mixing Lemma applying to r-partite weighted graphs.