Expanders and right-angled Artin groups

Abstract

The purpose of this paper is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs {Γi}i∈N forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups {A(Γi)}i∈N agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of vector space expanders, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky–Zelmanov and Bourgain–Yehudayoff.