Mathematics – Group Theory
Scientific paper
2010-10-28
J. Amer. Math. Soc. 24 (2011), 1-28
Mathematics
Group Theory
This article has been withdrawn by the author due to duplicate posting and can now be found at arXiv:0711.2704
Scientific paper
We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erd\H{o}s-R\'enyi random graphs. We find the threshold for simple connectivity to be p = n^{-1/2}. This is in contrast to the threshold for vanishing of the first homology group, which was shown earlier by Linial and Meshulam to be p = 2 log(n)/n. We use a variant of Gromov's local-to-global theorem for linear isoperimetric inequalities to show that when p = O(n^{-1/2 -\epsilon}) the fundamental group is word hyperbolic. Along the way we classify the homotopy types of sparse 2-dimensional simplicial complexes and establish isoperimetric inequalities for such complexes.
Babson Eric
Hoffman Christopher
Kahle Matthew
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