Systolic Expanders of Every Dimension

In recent years a high dimensional theory of expanders has emerged. The notion of combinatorial expanders of graphs (i.e. the Cheeger constant of a graph) has seen two generalizations to high dimensional simplicial complexes. One generalization, known as coboundary expansion is due to Linial and Meshulem; the other, which we term here systolic expansion, is due to Gromov, who showed that systolic expanders have the topological overlapping property. No construction (either random or explicit) of bounded degree combinational expanders (according to either definition) were known until a recent work of [KKL]. The work of [KKL] provided the first bounded degree systolic expanders of dimension two. No bounded degree combinatorial expanders are known in higher dimensions. In this work we show explicit bounded degree systolic expanders of every dimension. This solves affirmatively an open question raised by Gromov [G], who asked whether there exists bounded degree simplicial complexes with the topological overlapping property in every dimension. Moreover, we provide a local to global criteria on a complex that implies systolic expansion: Namely, if the 1-skeleton graph underlying a d-dimensional complex is a good expander graph and all its links are both coboundary expanders and good expander graphs, then the (d− 1)dimensional skeleton of the complex is a systolic expander.