The fundamental group of random 2-complexes

Abstract

We study Linial-Meshulam random 2-complexes, which are two-dimensional analogues of Erdős-Rényi 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 -ε}) 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.

This article has been withdrawn by the author due to duplicate posting and can now be found at arXiv:0711.2704

Cited by in corpus (4)

• Topology of random 2-complexes
• Finite Volume Spaces and Sparsification
• The expected genus of a random chord diagram
• The threshold function for vanishing of the top homology group of random $d$-complexes