A sharp threshold for Kazhdan's property (T)

Details A sharp threshold for Kazhdan's property (T) A sharp threshold for Kazhdan's property (T)

2012-01-02

arxiv.org/abs/1201.0425v1

Mathematics

Combinatorics

27 pages

Scientific paper

We show that $p = 2\log{n} / n$ is a sharp threshold for the fundamental group of random 2-complexes to have Kazhdan's property (T). This concurs with the threshold for the vanishing of the first homology, found earlier by Linial and Meshulam. Establishing this threshold requires new estimates and concentration results for the spectral gap of Erd\H{o}s-Renyi random graphs near the connectivity threshold, which are of independent interest.

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