High-dimensional expanders from Kac–Moody–Steinberg groups

Abstract

High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by the work of Kaufman and Oppenheim, we use coset complexes over quotients of Kac–Moody–Steinberg groups of rank $d+1$, $d$-spherical and purely $d$-spherical. We prove that infinite families of such quotients exist provided that the underlying field is of size at least 4 and the Kac–Moody–Steinberg group is 2-spherical, giving rise to new families of bounded degree high-dimensional expanders. In case the generalized Cartan matrix we consider is affine, we recover the construction of O’Donnell and Pratt from 2022 (and thus also the one by Kaufman and Oppenheim) by considering Chevalley groups as quotients of affine Kac–Moody–Steinberg groups. Moreover, our construction applies to the case where the root system is of type ${\tilde{G}}_2$, a case that was not covered in earlier works.