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Freeness of the random fundamental group

Department of Mathematics, The Ohio State University, USA

https://doi.org/10.1142/S1793525319500468Cited by:3 (Source: Crossref)

Abstract

Let $Y (n, p)$ $Y (n, p)$ denote the probability space of random 2-dimensional simplicial complexes in the Linial–Meshulam model, and let $Y \sim Y (n, p)$ $Y \sim Y (n, p)$ denote a random complex chosen according to this distribution. In a paper of Cohen, Costa, Farber and Kappeler, it is shown that for $p = o (1 / n)$ $p = o (1 / n)$ with high probability $π_{1} (Y)$ $π_{1} (Y)$ is free. Following that, a paper of Costa and Farber shows that for values of $p$ $p$ which satisfy $3 / n < p ≪ n^{- 46 / 47}$ $3 / n < p ≪ n^{- 46 / 47}$ with high probability, $π_{1} (Y)$ $π_{1} (Y)$ is not free. Here, we improve on both these results to show that there are explicit constants $γ_{2} < c_{2} < 3$ $γ_{2} < c_{2} < 3$ , so that for $p < γ_{2} / n$ $p < γ_{2} / n$ with high probability $Y$ $Y$ has free fundamental group and that for $p > c_{2} / n$ $p > c_{2} / n$ with high probability $Y$ $Y$ has fundamental group which either is not free or is trivial.

Keywords:

AMSC: 60C05, 20J06

References

1. L. Aronshtam and N. Linial, When does top homology in a random simplicial complex vanish?, Random Struct. Algorithms 46 (2013) 26–35. Crossref, Google Scholar
2. L. Aronshtam and N. Linial, The threshold for d-collapsibility in random complexes, Random Struct. Algorithms 48 (2015) 260–269. Crossref, Google Scholar
3. L. Aronshtam, N. Linial, T. Luczak and R. Meshulam, Collapsibility and vanishing of top homology in random simplicial complexes, Discrete Comput. Geom. 49 (2013) 317–334. Crossref, Google Scholar
4. E. Babson, C. Hoffman and M. Kahle, The fundamental group of random 2-complexes, J. Am. Math. Soc. 24 (2011) 1–28. Crossref, Google Scholar
5. K. S. Brown, Cohomology of Groups (Springer-Verlag, 1982). Crossref, Google Scholar
6. D. Cohen, A. Costa, M. Farber and T. Kappeler, Topology of random 2-complexes, Discrete Comput. Geom. 47 (2012) 117–149. Crossref, Google Scholar
7. A. E. Costa and M. Farber, The asphericity of random 2-dimensional complexes, Random Struct. Algorithms 46 (2015) 261–273. Crossref, Google Scholar
8. A. E. Costa and M. Farber, Geometry and topology of random 2-complexes, Isr. J. Math. 209 (2015) 883–927. Crossref, Google Scholar
9. C. Hoffman, M. Kahle and E. Paquette, Spectral gaps of random graphs and applications to random topology, arXiv:1201.0425. Google Scholar
10. M. Kahle, F. H. Lutz, A. Newman and K. Parsons, Cohen–Lenstra heuristics for torsion in homology of random complexes, arXiv:1710.05683. Google Scholar
11. N. Linial and R. Meshulam, Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487. Crossref, Google Scholar
12. N. Linial and Y. Peled, On the phase transition in random simplicial complexes, Ann. Math. 184 (2016) 745–773. Crossref, Google Scholar
13. N. Linial and Y. Peled, Random simplicial complexes: Around the phase transition, in A Journey Through Discrete Mathematics (Springer, 2017), pp. 543–570. Crossref, Google Scholar
14. T. Luczak and Y. Peled, Integral homology of random simplicial complexes, Discrete Comput. Geom. 59 (2018) 131–142. Crossref, Google Scholar
15. R. Meshulam and N. Wallach, Homological connectivity of random $k$ $k$ -dimensional complexes, Random Struct. Algorithms 34 (2009) 408–417. Crossref, Google Scholar