Isoperimetric inequalities for Ramanujan complexes and topological expanders

Alex Lubotsky (Hebrew U. and NYU)

Abstract:
Expander graphs have been intensively studied in the last four decades. In recent years a high dimensional theory of expanders has emerged, and several generalizations have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, due to Gromov , is whether bounded degree high dimensional expanders exist for d>1.

We present an explicit construction of bounded degree complexes of dimension d= 2 which are topological expanders, thus answering Gromov’s question in the affirmative for this case. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes.

All notions will be explained.

Joint work with T. Kaufman and D. Kazhdan.

Wednesday, Oct. 01, 4:30 – 5:30 p.m.
Mathematics 520
Tea will be served at 4:00 p.m.

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