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 . We present an explicit construction of bounded degree complexes of dimension 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 systolic invariants of these complexes, which seem to be the first linear systolic bounds. The expansion results are deduced from these isoperimetric inequalities.