## The fundamental group of random $2$-complexes

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- by Eric Babson, Christopher Hoffman and Matthew Kahle
- J. Amer. Math. Soc.
**24**(2011), 1-28 - DOI: https://doi.org/10.1090/S0894-0347-2010-00677-7
- Published electronically: August 30, 2010
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## Abstract:

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 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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## Bibliographic Information

**Eric Babson**- Affiliation: Department of Mathematics, University of California at Davis, Davis, California 95616
- Email: babson@math.ucdavis.edu
**Christopher Hoffman**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 634876
- Email: hoffman@math.washington.edu
**Matthew Kahle**- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- Email: kahle@math.stanford.edu
- Received by editor(s): November 7, 2008
- Received by editor(s) in revised form: July 9, 2010
- Published electronically: August 30, 2010
- Additional Notes: The second author was supported in part by NSA grant #H98230-05-1-0053 and NSF grant #DMS-0501102 and by an AMS Centennial Fellowship.

The third author was supported in part by the University of Washington’s NSF-VIGRE grant #DMS-0354131.

We would also like to thank MSRI and the Institute for Advanced Studies at the Hebrew University of Jerusalem where some of the research was done. - © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**24**(2011), 1-28 - MSC (2010): Primary 20F65; Secondary 05C80
- DOI: https://doi.org/10.1090/S0894-0347-2010-00677-7
- MathSciNet review: 2726597