## The fundamental group of random 2-complexes.(English)Zbl 1270.20042

Summary: We study Linial-Meshulam random 2-complexes $$Y(n,p)$$, which are 2-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 N. Linial and R. Meshulam [Combinatorica 26, No. 4, 475-487 (2006; Zbl 1121.55013)] 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 $$p=O(n^{-1/2-\varepsilon})$$, 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.

### MSC:

 20F65 Geometric group theory 55U10 Simplicial sets and complexes in algebraic topology 05C80 Random graphs (graph-theoretic aspects) 20F67 Hyperbolic groups and nonpositively curved groups 05E45 Combinatorial aspects of simplicial complexes 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 57M07 Topological methods in group theory

Zbl 1121.55013
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### References:

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