Bounded degree cosystolic expanders of every dimension

Abstract

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.

High Dimensional Expanders

“…What is known is a somewhat weaker result which still suffices to answer, in the affirmative, Gromov's question on the existence of bounded degree topological expanders. The following theorem was proved by Kaufman-Kazhdan-Lubotzky [KKL16] for d ≤ 3 and by Evra and Kaufman [EvKa16] for general d. As this Theorem holds for every d, it solves Gromov's problem, but in a somewhat unexpected way. We do believe that X in the theorem are also cosystolic expanders and not just Y .…”

“…In fact, as of now, there is no known "random model" for ddimensional simplicial complexes of bounded degree (in the strong sense -see below) which gives high dimensional topological expanders. This is surprising as the existence of such topological expanders is known by now by ([KKL16], [EvKa16]) as was explained in §3. One may start to wonder if such a model exists at all, or maybe topological bounded degree expanders of high dimension are very rare objects.…”

Untitled

“…Remark 1.9. We note that the notion of hypergraph expansion is not directly related to the notion of coboundary expansion of simplicial complexes [15,14,11]. Aside from an obvious distinction that the former applies to hypergraphs and the latter applies to simplicial complexes, these two types of expansion measure very different parameters of hypergraphs/simplicial complexes.…”

Expander Graphs – Both Local and Global

“…Several different ways were proposed to quantify this notion. For the definitions of cosystolic and coboundary expansion, see e.g., [EK16]. 5.1.…”

“…Note that this is the only family of Ramanujan complexes whose 1-skeleton is (a, b)-regular for some a and b. These graphs have many high dimensional expansion properties, see e.g., [DK17] and [EK16]. (3) An additional group theoretic construction is due to Kaufman and Oppenheim in [KO17] (The 2-dimensional case).…”