Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006 ; Lubotzky, Bull Am Math Soc, 49:113–162, 2012 ). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion . It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416–526, 2010 ), is whether bounded degree high dimensional expanders exist for d ≥ 2 . We present an explicit construction of bounded degree complexes of dimension d = 2 which are topological expanders , thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders . The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on 𝔽_2 systolic invariants of these complexes, which seem to be the first linear 𝔽_2 systolic bounds. The expansion results are deduced from these isoperimetric inequalities.