The fundamental group of random 2-complexes

Details The fundamental group of random 2-complexes The fundamental group of random 2-complexes

2007-11-16

arxiv.org/abs/0711.2704v4

J. Amer. Math. Soc. 24 (2011), 1-28

Mathematics

Combinatorics

28 pages, 3 figures; major revisions

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. These intermediate results do not involve randomness and may be of independent interest.

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