In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.A family of d-dimensional simplicial complexes is said to have the topological overlapping property, if there exists c > 0, such that each member of the family has the c-topological overlapping property.Gromov then proved the remarkable result that, for a fixed d ∈ N, the family of complete ddimensional complexes have the topological overlapping property. This result is a very striking generalization of classical results from convex combinatorics due to Boros-Furedi [BF] (for d = 2) * Hebrew University, ISRAEL.

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Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T ) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have 'high girth' and high chromatic number, generalizing a well known result about Ramanujan graphs.

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