The fundamental group of random 2-complexes
Abstract
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.
Keywords
Cite
@article{arxiv.1010.6043,
title = {The fundamental group of random 2-complexes},
author = {Eric Babson and Christopher Hoffman and Matthew Kahle},
journal= {arXiv preprint arXiv:1010.6043},
year = {2011}
}
Comments
This article has been withdrawn by the author due to duplicate posting and can now be found at arXiv:0711.2704