Gromov and independently Linial and Meshulam suggested two related notions of high dimensional expanders. Gromov defined "topological overlapping" while Linial and Meshulam defined "coboundary expansion". A major open problem (asked by Gromov and others) is whether *bounded degree* high dimensional expanders according to these definitions exist for $d \geq 2$. We show, for the first time, bounded degree complexes of dimension $d=2$ which have the topological overlapping property. Assuming a conjecture of Serre on the congruence subgroup property, these complexes are also coboundary expanders.
Joint work with David Kazhdan and Alex Lubotzky.