Bounded Degree High Dimensional Expanders and Topological Overlapping

Expander graphs are widely studied, and various methods are known to obtain bounded degree expander graphs. Recently, there is a growing interest in understanding combinational expansion in higher dimensions (higher dimensional simplicial complexes). However, bounded degree combinatorial expanders (random or explicit) were not known till our work. We present a local to global criterion on a complex that implies combinatorial expansion. We use our criterion to present explicit bounded degree high dimensional expanders. This solves in the affirmative an open question raised by Gromov, who asked whether bounded degree high dimensional expanders could at all exist. Gromov has shown that high dimensional expansion of a $d$-dimensional complex, implies that the complex has a topological overlapping property; Namely, every continuous mapping of such a complex to $R^d$ must have a point in $R^d$ that is covered by a constant fraction of the topological $d$-simplices obtained by the mapping. Gromov has asked whether there exist bounded degree complexes with the topological overlapping property. Our work provides an affirmative answer to this question. Based on joint works with David Kazhdan and Alex Lubotzky, and with Shai Evra.