Random Steiner systems and bounded degree coboundary expanders of every dimension

Abstract

We introduce a new model of random d-dimensional simplicial complexes, for d ≥ 2, whose (d − 1)-cells have bounded degrees. We show that with high probability, complexes sampled according to this model are coboundary expanders. The construction relies on Keevash’s recent result on designs (The existence of designs; arXiv:1401.3665, 2014), and the proof of the expansion uses techniques developed by Evra and Kaufman in (Bounded degree cosystolic expanders of every dimension; arXiv:1510.00839, 2015). This gives a full solution to a question raised in Dotterrer and Kahle (J Topol Anal 4(4): 499–514, 2012), which was solved in the two-dimensional case by Lubotzky and Meshulam (Adv Math 272: 743–760, 2015).