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The fundamental group of random $ 2$-complexes


Authors: Eric Babson, Christopher Hoffman and Matthew Kahle
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
Published electronically: August 30, 2010
MathSciNet review: 2726597
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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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Additional 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
Email: hoffman@math.washington.edu

Matthew Kahle
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: kahle@math.stanford.edu

DOI: https://doi.org/10.1090/S0894-0347-2010-00677-7
Keywords: Random groups, hyperbolic groups
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.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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